Convergence Tests

Strategy for Series Convergence Tests https://eddieguo.ca/assets/downloads/notes/math101.pdf

Cauchy’s Convergence Test

• (5.4) The series $\sum a_n$ converges if and only if for every $\epsilon >0$ there exists $N_{\epsilon }$ such that if $\forall n>N_{\epsilon }$, and $\forall p\in \mathbb{N}$ we have $\left|s_{n+p}-s_n\right|=\left|a_{n+1}+a_{n+2}+\cdots +a_{n+p}\right|<\epsilon$.

Divergence Test

• (5.5) If ${\sum }_{n=1}^{\infty }{a}_n$ converges, then $\underset{n\to \infty }{\lim }{a}_n=0$.
• (The Divergence Test) If $\underset{n\to \infty }{\lim }{a}_n\ne 0$, then ${\sum }_{n=1}^{\infty }{a}_n$ diverges.

p-Test

• (e5.8) The p-series $\sum _{n=1}^{\infty }\frac{1}{n^p}$ converges if $p>1$ and diverges if $p\le 1$

Absolute Convergence Test

• (5.6) If the seires $\sum \left|a_n\right|$ converges, then $\sum a_n$ converges as well.
• see also Convergence Absolutely

Monotone Converges Theorem for Series

• (5.13 MCT) $(a_n)\ge 0⟺(s_n)$ is increasing. In this case:
  • $\sum a_n$ converges $⟺$ $\sum a_n$ is bounded $⟺$ $(s_n)$ is bounded
    • In this case, $\sum a_n=\lim s_n=\sup \{s_n:n\in \mathbb{N}\}$
  • The series $\sum a_n$ diverges to infinity $⟺$ $(s_n)$ is unbounded.
• $(a_n)>0⟺(s_n)$ is strictly increasing.
• $(a_n)\le 0⟺(s_n)$ is decreasing.
• $(a_n)<0⟺(s_n)$ is strictly decreasing.

Comparison test

Given series $\sum a_n$ and $\sum b_n$.

Direct Comparison test

• The Comparison Test (5.14, “first”) If $0\le a_n\le b_n$ for all $n>N$, then
  • If $\sum b_n$ converges, then $\sum a_n$ converges.
  • If $\sum a_n$ diverges, then $\sum b_n$ diverges. (they both diverge to infinity)

Limit Comparison Test

Given $a_n\ge 0$ and $b_n\ge 0$ for all $n>N$.

• (5.15, “second”) If $\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=c$, where $c$ is a positive real number, then
  • The series $\sum a_n$ converges, if and only if the series $\sum b_n$ converges.
• (q5.21)
  • If $\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=0$, and $\sum b_n$ converges, then $\sum a_n$ converges.
  • If $\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=\infty$, and $\sum b_n$ diverges, then $\sum a_n$ diverges.

Root test (Cauchy)

• (5.16) Root test (Cauchy)

  • (a.) If there exists $q<1$ such that $\sqrt[n]{|a_n|}\le q$ for almost all $n$, then the series $\sum a_n$ converges absolutely.
  • (b.) If $\sqrt[n]{|a_n|}\ge 1$ for infinitely many $n$, then the series $\sum a_n$ diverges.

• (5.16*) Given $c=\underset{n\to \infty }{\mathrm{limsup}}\sqrt[n]{|a_n|}$. or ($c=\underset{n\to \infty }{\lim }\sqrt[n]{|a_n|}$ exists (5.16**))

  • If $c<1$, then the series $\sum a_n$ converges absolutely.
  • If $c>1$, then the series $\sum a_n$ diverges.
  • If $c=1$, then the test is inconclusive.

Ratio test (d’Alembert)

• Ratio test (d’Alembert) (given $a_n\ne 0$ for all $n$)
  • (5.17)
    • (a.) If there exists $q<1$ such that $\left|\frac{a_{n+1}}{a_n}\right|\le q$ for almost all $n$, then the series $\sum a_n$ converges absolutely.
    • (b.) If $\left|\frac{a_{n+1}}{a_n}\right|\ge 1$ for almost all $n$, then the series $\sum a_n$ diverges.
  • (5.17**) Given $c=\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|$.
    • If $c<1$, then the series $\sum a_n$ converges absolutely.
    • If $c>1$, then the series $\sum a_n$ diverges.
    • If $c=1$, then the test is inconclusive. (q5.26) If there exists $q<1$ such that $\left|\frac{a_{n+1}}{a_n}\right|\le q<1$ for almost all $n$, then there exists $q\le q^{\prime }<1$ such that $\sqrt[n]{|a_n|}\le q^{\prime }$ for almost all $n$.

If we found that $\sum a_n$ diverges by either ratio test or root test, then $\underset{n\to \infty }{\lim }{a}_n\ne 0$.

Cauchy Condensation Test

• (5.18) Cauchy condensation test - Let $(a_n)$ be a decreasing sequence of nonnegative terms. The series ${\sum }_{n=1}^{\infty }{a}_n$ converges if and only if the series ${\sum }_{n=1}^{\infty }{2}^n{a}_{2^n}$ converges.

Integral Test

(5.19)

• Given a series $\sum a_k$ of nonnegative terms, where $a_k$ is decreasing.
• Given $f(x)$ is decreasing (weakly), nonnegative on $[1,\infty )$, and integrable every fintie interval. And $a_n=f(n)$ for all $n\in \mathbb{N}$.

The series $\sum a_k$ converges if and only if the improper integral ${\int }_1^{\infty }f(x)dx$ converges.

We can use integral test also when $f$ is decreasing and nonnegative on $[k_0,\infty )$ for some $k_0\in \mathbb{N}$. In this case, the series ${\sum }_{k=k_0}^{\infty }{a}_k$ converges if and only if the improper integral ${\int }_{k_0}^{\infty }f(x)dx$ converges. Hence, the series ${\sum }_{k=1}^{\infty }{a}_k$ converges if and only if the series ${\sum }_{k=k_0}^{\infty }{a}_k$ converges.

Alternating Series Test (Leibniz)

• (5.20) Let $(a_n)$ be a decreasing, null, (thus nonnegative) sequence. Then:
  • (A.) The series ${\sum }_{n=1}^{\infty }(-1{)}^{n+1}{a}_n$ converges. (This is a special case of Dirichlet’s test. see q5.32)
  • Alternating Series Estimation Theorem - If $S=\sum (-1{)}^{n+1}{a}_n$, and $S_n$ is the $n$-th partial sum, then for all $n$ we have:
    • (B.) $S$ is between $S_n$ and $S_{n+1}$.
    • (C.) The error $|S-S_n|$ is less than $a_{n+1}$.

A series of the form $\sum (-1{)}^{n+1}{a}_n$ is called an alternating series

These conclusions hold also for the series $\sum (-1{)}^n{a}_n$.

Dirichlet’s Test

• (5.22) Let $\sum a_k$ be a bounded series, and $(b_k)$ be a monotone null sequence. Then the series $\sum a_k{b}_k$ converges.

Abel’s Test

• (5.23) Let $\sum a_k$ be a convergent series, and let $(b_k)$ be a sequence that is bounded and monotone. Then the series $\sum a_k{b}_k$ converges.