Convergence tests

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Infinite Series Fundamentals

An infinite series is an expression of the form $\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \ldots$, where $a_n$ is the general term of the sequence.

Sequence of Partial Sums

For a series $\sum_{n=1}^{\infty} a_n$, the sequence of partial sums ${S_N}$ is defined as: \(S_N = \sum_{n=1}^{N} a_n = a_1 + a_2 + \ldots + a_N\)

Convergence Definition

A series $\sum_{n=1}^{\infty} a_n$ converges if the sequence of partial sums ${S_N}$ converges to a finite limit $S$. That is: \(\lim_{N \to \infty} S_N = S\)

If this limit doesn’t exist or is infinite, the series diverges.

Necessary Condition for Convergence

If $\sum_{n=1}^{\infty} a_n$ converges, then $\lim_{n \to \infty} a_n = 0$.

Note: This is a necessary but not sufficient condition. If $\lim_{n \to \infty} a_n \neq 0$, the series diverges. However, if $\lim_{n \to \infty} a_n = 0$, the series may or may not converge.

Convergence Tests for Series with Positive Terms

1. Comparison Test

If $0 \leq a_n \leq b_n$ for all $n \geq N$ (some fixed $N$), then:

  • If $\sum b_n$ converges, then $\sum a_n$ converges.
  • If $\sum a_n$ diverges, then $\sum b_n$ diverges.

2. Limit Comparison Test

If $a_n > 0$, $b_n > 0$ for all sufficiently large $n$, and $\lim_{n \to \infty} \frac{a_n}{b_n} = L$ where $0 < L < \infty$, then:

  • $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

3. Ratio Test

For a series $\sum a_n$ with positive terms, compute $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L$:

  • If $L < 1$, the series converges absolutely.
  • If $L > 1$ or $L = \infty$, the series diverges.
  • If $L = 1$, the test is inconclusive.

4. Root Test

For a series $\sum a_n$ with positive terms, compute $\lim_{n \to \infty} \sqrt[n]{a_n} = L$:

  • If $L < 1$, the series converges absolutely.
  • If $L > 1$ or $L = \infty$, the series diverges.
  • If $L = 1$, the test is inconclusive.

5. Integral Test

If $f(x)$ is a positive, continuous, decreasing function for $x \geq k$ (for some $k$), and $f(n) = a_n$ for integers $n \geq k$, then:

  • $\sum_{n=k}^{\infty} a_n$ converges if and only if $\int_{k}^{\infty} f(x) \, dx$ converges.

6. p-Series Test

The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$:

  • Converges if $p > 1$
  • Diverges if $p \leq 1$

7. Cauchy Condensation Test

If ${a_n}$ is a positive, non-increasing sequence, then:

  • $\sum_{n=1}^{\infty} a_n$ converges if and only if $\sum_{n=0}^{\infty} 2^n a_{2^n}$ converges.

Tests for Series with Terms of Any Sign

1. Absolute Convergence Test

If $\sum_{n=1}^{\infty} |a_n|$ converges, then $\sum_{n=1}^{\infty} a_n$ also converges, and the series is said to be absolutely convergent.

2. Alternating Series Test (Leibniz’s Test)

If ${a_n}$ is a sequence such that:

  • $a_n > 0$ for all $n$
  • $a_{n+1} \leq a_n$ for all $n$ (non-increasing)
  • $\lim_{n \to \infty} a_n = 0$

Then the alternating series $\sum_{n=1}^{\infty} (-1)^{n+1} a_n$ converges.

3. Dirichlet’s Test

If ${a_n}$ is a monotonic sequence converging to 0, and the partial sums of ${b_n}$ are bounded (i.e., $|\sum_{j=1}^{n} b_j| \leq M$ for all $n$ and some constant $M$), then: \(\sum_{n=1}^{\infty} a_n b_n \text{ converges}\)

4. Abel’s Test

If $\sum_{n=1}^{\infty} b_n$ converges and ${a_n}$ is a monotonic bounded sequence, then: \(\sum_{n=1}^{\infty} a_n b_n \text{ converges}\)

Special Series and Their Convergence

Geometric Series

\(\sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + ar^3 + \ldots\)

  • Converges to $\frac{a}{1-r}$ when $ r < 1$
  • Diverges when $ r \geq 1$

Harmonic Series

\(\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \ldots\)

  • Diverges (p = 1)

p-Series

\(\sum_{n=1}^{\infty} \frac{1}{n^p} = 1 + \frac{1}{2^p} + \frac{1}{3^p} + \ldots\)

  • Converges for $p > 1$
  • Diverges for $p \leq 1$

Alternating Harmonic Series

\(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots\)

  • Converges to $\ln 2$ (by Leibniz’s test)

Asymptotic Comparison for Complex Series

When dealing with complex series, it’s often useful to compare with a known series using asymptotic comparison:

Asymptotic Comparison Test

If $\lim_{n \to \infty} \frac{a_n}{b_n} = L$ where $0 < L < \infty$, then:

  • $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

This is similar to the limit comparison test but is often used with more complex terms where direct comparison is difficult.

Worked Examples

Example 1: Ratio Test for Factorial Series

Determine if the series $\sum_{n=1}^{\infty} \frac{2^n}{n!}$ converges.

Solution: Using the ratio test: \(\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{2^{n+1}/(n+1)!}{2^n/n!} = \lim_{n \to \infty} \frac{2^{n+1}}{2^n} \cdot \frac{n!}{(n+1)!} = \lim_{n \to \infty} 2 \cdot \frac{1}{n+1} = 0\)

Since the limit is 0 (which is < 1), the series converges absolutely.

Example 2: Comparison Test with Square Root

Establish if the series $\sum_{n=1}^{\infty} (\sqrt{n+1} - \sqrt{n})$ converges.

Solution: First, rationalize to find an equivalent form: \(\sqrt{n+1} - \sqrt{n} = \frac{(\sqrt{n+1} - \sqrt{n})(\sqrt{n+1} + \sqrt{n})}{(\sqrt{n+1} + \sqrt{n})} = \frac{(n+1) - n}{(\sqrt{n+1} + \sqrt{n})} = \frac{1}{\sqrt{n+1} + \sqrt{n}}\)

As $n \to \infty$, $\sqrt{n+1} + \sqrt{n} \sim 2\sqrt{n}$, so: \(\sqrt{n+1} - \sqrt{n} \sim \frac{1}{2\sqrt{n}}\)

Compare with $\frac{1}{\sqrt{n}}$ using the limit comparison test: \(\lim_{n \to \infty} \frac{\sqrt{n+1} - \sqrt{n}}{\frac{1}{\sqrt{n}}} = \lim_{n \to \infty} \frac{\sqrt{n}}{\sqrt{n+1} + \sqrt{n}} = \lim_{n \to \infty} \frac{1}{1 + \sqrt{\frac{n+1}{n}}} = \frac{1}{2}\)

Since $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ diverges (p-series with p = 1/2 < 1), and the limit is a finite positive number, the original series also diverges.

Example 3: Alternating Series

Establish if the series $\sum_{k=2}^{\infty} \frac{(-1)^k k^2}{k^3-2}$ converges.

Solution: Step 1: Check if the terms approach zero \(\lim_{k \to \infty} \frac{k^2}{k^3-2} = \lim_{k \to \infty} \frac{k^2}{k^3(1-\frac{2}{k^3})} = \lim_{k \to \infty} \frac{1}{k} = 0\)

Step 2: Check for absolute convergence For absolute convergence, consider $\sum_{k=2}^{\infty} \frac{k^2}{k^3-2}$

As $k \to \infty$, $\frac{k^2}{k^3-2} \sim \frac{k^2}{k^3} = \frac{1}{k}$

Using the limit comparison test with the harmonic series: \(\lim_{k \to \infty} \frac{\frac{k^2}{k^3-2}}{\frac{1}{k}} = \lim_{k \to \infty} \frac{k \cdot k^2}{k^3-2} = \lim_{k \to \infty} \frac{k^3}{k^3-2} = 1\)

Since the harmonic series diverges, the series $\sum_{k=2}^{\infty} \frac{k^2}{k^3-2}$ also diverges, meaning the original alternating series is not absolutely convergent.

Step 3: Apply the alternating series test To check if the series is conditionally convergent, we need to verify if $\frac{k^2}{k^3-2}$ is decreasing for $k \geq 2$.

The derivative of $f(x) = \frac{x^2}{x^3-2}$ is: \(f'(x) = \frac{2x(x^3-2) - 3x^2 \cdot x^2}{(x^3-2)^2} = \frac{2x^4-4x-3x^4}{(x^3-2)^2} = \frac{-x^4-4x}{(x^3-2)^2}\)

For $x > 0$, this derivative is negative, indicating that $f(x)$ is decreasing for $k \geq 2$.

Since the terms approach zero and form a decreasing positive sequence, by the alternating series test, the series $\sum_{k=2}^{\infty} \frac{(-1)^k k^2}{k^3-2}$ converges.

Therefore, the series is conditionally convergent.

Example 4: Ratio Test for Complex Expression

Determine if the series $\sum_{n=1}^{\infty} \frac{n^2}{2^n}$ converges.

Solution: Using the ratio test: \(\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{(n+1)^2/2^{n+1}}{n^2/2^n} = \lim_{n \to \infty} \frac{(n+1)^2}{n^2} \cdot \frac{1}{2} = \lim_{n \to \infty} \frac{1}{2} \cdot \left(1 + \frac{1}{n}\right)^2 = \frac{1}{2}\)

Since the limit is 1/2 (< 1), the series converges absolutely.

Example 5: Testing Series with Logarithm

Establish if the series $\sum_{n=1}^{\infty} \frac{1}{n} - \ln\left(1+\frac{1}{n}\right)$ converges.

Solution: Step 1: Use the Taylor expansion of $\ln(1+x)$ around $x = 0$: \(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \ldots\)

Substituting $x = \frac{1}{n}$: \(\ln\left(1+\frac{1}{n}\right) = \frac{1}{n} - \frac{1}{2n^2} + \frac{1}{3n^3} - \ldots\)

Step 2: Find the difference: \(\frac{1}{n} - \ln\left(1+\frac{1}{n}\right) = \frac{1}{n} - \left(\frac{1}{n} - \frac{1}{2n^2} + \frac{1}{3n^3} - \ldots\right) = \frac{1}{2n^2} - \frac{1}{3n^3} + \ldots\)

Step 3: Compare with a known convergent series For large $n$, the dominant term is $\frac{1}{2n^2}$, which can be compared with $\frac{1}{n^2}$.

\[\lim_{n \to \infty} \frac{\frac{1}{n} - \ln\left(1+\frac{1}{n}\right)}{\frac{1}{n^2}} = \lim_{n \to \infty} \frac{\frac{1}{2n^2} - \frac{1}{3n^3} + \ldots}{\frac{1}{n^2}} = \frac{1}{2}\]

Since $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges (p-series with p = 2 > 1), and the limit is a finite positive number, the original series also converges.

Strategic Approach to Convergence Problems

  1. Examine the General Term:
    • Does $a_n \to 0$ as $n \to \infty$? If not, the series diverges.
    • Is $a_n$ always positive, always negative, or does it change sign?
    • Does $a_n$ involve factorial, exponential, or power functions?
  2. Choose the Appropriate Test:
    • For positive terms, try ratio test for factorials, exponentials
    • For alternating series, check Leibniz’s test conditions
    • For complex expressions, try asymptotic comparison
    • For rational functions, compare with p-series
  3. When Tests are Inconclusive:
    • If one test is inconclusive, try another
    • For series with ratio = 1, try the root test or direct comparison
    • Break the series into parts if needed
  4. For Special Cases:
    • Recognize geometric series: $\sum r^n$ converges when $ r < 1$
    • Identify p-series: $\sum \frac{1}{n^p}$ converges when $p > 1$
    • Look for telescoping series where most terms cancel
  5. Verification:
    • Double-check your work, especially with complex manipulations
    • Confirm that all conditions of the test you’re using are satisfied

See:

  1. Consider the series $\sum_{n=1}^{\infty} \sqrt{n+1} - \sqrt{n}$ a) Verify that $a_n = \sqrt{n+1} - \sqrt{n}$ satisfies $\lim_{n \to \infty} a_n = 0$ b) Prove that the series diverges.

  2. Consider the series $\sum_{n=1}^{\infty} \frac{(n^2 + 1)}{2^n}$ a) Verify that $a_n = \frac{(n^2+1)}{2^n}$ satisfies $\lim_{n \to \infty} a_n = 0$ b) Prove that the series converges by using the ratio test.

  3. Consider the series $\sum_{n=1}^{\infty} \frac{(n^2 + 1)}{n^2\sqrt{n+1}}$ a) Verify that $a_n = \frac{(n^2+1)}{n^2\sqrt{n+1}}$ satisfies $\lim_{n \to \infty} a_n = 0$ b) Prove that the series diverges by using an asymptotic comparison with $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$.

  4. (Written Test, Analysis - TEST 2) Given the series $\sum_{k=2}^{\infty} a_k$ with $a_k = \frac{k^2-\sqrt{k}}{k^3-2k}$ a) Establish if $a_k$ tends to zero as $k$ tends to infinity. b) Establish if the series is absolutely convergent, convergent, divergent or not convergent.

  5. Consider the series $\sum_{n=1}^{\infty}\left(1-\cos\left(\frac{1}{n}\right)\right)$. Mark true or false in the following sentences: a) The series is absolutely convergent. b) The series is convergent. c) The series has the same character of the series $\sum_{n=1}^{\infty}\frac{1}{n^2}$ d) The series diverges.

  6. Study the character of the infinite series $\sum_{n=1}^{\infty} n \ln\left(1 + \frac{1}{n^3}\right)$.

  7. Given the series $\sum_{k=1}^{+\infty} \frac{k^2-1}{k^4+3k^2+k-1}$ a) Establish if the series has positive terms. b) Prove that the series diverges.

  8. Given the series $\sum_{k=2}^{+\infty} a_k$ with $a_k = \frac{k \ln k}{k^2+2}$ a) Establish if $a_k$ tends to zero as $k$ tends to infinity. b) Establish if the series is absolutely convergent, convergent, divergent or not convergent.

  9. Given the series $\sum_{k=1}^{+\infty} \frac{k^2-1}{k^{3/2}+k}$ a) Establish if the series has positive terms. b) Prove that the series diverges.

  10. Given the series $\sum_{k=1}^{+\infty} (-1)^k \frac{k^2}{k^4+1}$ a) Establish if the series has positive terms for every $k$. b) Establish if the series is absolutely convergent, convergent, divergent or not convergent.

  11. TEST 2, Question 3: Establish if the series $\sum_{k=2}^{\infty} \frac{(-1)^k k^2}{k^3-2}$ a) Has positive terms for every $k$. b) Is absolutely convergent, convergent, divergent or not convergent.

  12. Written test 2022-4-18, Question 1: Analyze the convergence of $\sum_{m=1}^{\infty} \frac{x^m}{m}(1+\frac{1}{m})$ using the asymptotic comparison test and the ratio test.

  13. Establish if the following sequences are convergent, divergent or indeterminate: a) $b_n = \frac{1}{1+\sqrt{n}}$ b) $c_n = \frac{2n-1}{3n-1}$ c) $d_n = \frac{6n^3-n^2+1}{n^3+5}$ d) $s_n = \sqrt{n^2+2n-1} - \sqrt{n^2+2}$ e) $t_n = \sqrt{n^2+1} - n$ f) $u_n = (\sqrt{n+1}-\sqrt{n+2})\sqrt{n}$ g) $v_n = \sqrt[3]{n} - \sqrt[3]{n+1}$ h) $x_n = \frac{n^2-1}{n+1} - \frac{1}{n}\frac{n^2-2}{3n+1}$ i) $y_n = \sqrt{n} - n$ j) $z_n = \frac{\sqrt{n^2+n+1}}{\sqrt{n+1}}$

  14. Given the series $\sum_{k=1}^{+\infty} \frac{(-1)^k k^2}{2^k+1}$: a) Establish if the series has positive terms. b) Prove that the series absolutely converges.

  15. Given the series $\sum_{k=1}^{+\infty} \frac{k^2-1}{k^4+3k^2+k-1}$: a) Establish if the series has positive terms. b) Prove that the series diverges.

  16. Given the series $\sum_{k=1}^{+\infty} \frac{2^k}{2^k+1}$: a) Establish if the series has positive terms. b) Prove that the series diverges.

  17. Given the series $\sum_{k=1}^{+\infty} \frac{k!}{(k+2)!+1}$: a) Establish if the series has positive terms. b) Prove that the series converges.

  18. Establish if the series $\sum_{n=1}^{\infty} a_n$ with $a_n = \frac{2^n}{n^3+1}$ is absolutely convergent, convergent, divergent or not convergent.

  19. Study the absolute and simple convergence of the following series: a) $\sum_{n=1}^{\infty} \frac{1}{n} - \ln(1+\frac{1}{n})$ b) $\sum_{n=1}^{\infty} \frac{(-1)^n n^5}{2^n}$ c) $\sum_{n=1}^{\infty} (-1)^n \sin\frac{1}{n}$ d) $\sum_{n=1}^{\infty} \sqrt{n+3} - \sqrt{n}$ e) $\sum_{k=1}^{\infty} \frac{5k-1}{3k^2+2}$ f) $\sum_{n=1}^{\infty} \frac{n}{2^n}$ g) $\sum_{n=1}^{\infty} \frac{2^n}{n!}$ h) $\sum_{n=1}^{\infty} \frac{n}{2^n}$