6 Infinite Series and Convergence Tests Free Online FlashCards

When does an infinite series converge?

An infinite series converges when its sequence of partial sums approaches a finite limit; otherwise, it diverges.

What does the nth-term test establish?

If $\lim_{n\to\infty}a_n\neq0$, then $\sum a_n$ diverges. However, $a_n\to0$ alone does not guarantee convergence.

What is the convergence rule for a geometric series?

A geometric series $\sum_{n=0}^{\infty}ar^n$ converges exactly when $|r|<1$, and then its sum is $\frac{a}{1-r}$.

How is a telescoping series evaluated?

For $\sum_{n=1}^{\infty}(b_n-b_{n+1})$, the partial sum is $S_N=b_1-b_{N+1}$. If $b_{N+1}\to L$, the series sums to $b_1-L$.

What is the direct comparison test for convergence?

For nonnegative terms, if $0\le a_n\le b_n$ eventually and $\sum b_n$ converges, then $\sum a_n$ converges.

When does the limit comparison test apply?

If $a_n,b_n>0$ and $\lim_{n\to\infty}\frac{a_n}{b_n}=L$ with $0<L<\infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

What conditions are required for the integral test?

The integral test applies when $a_n=f(n)$, where $f$ is positive, continuous, and eventually decreasing. Then $\sum a_n$ and $\int_N^{\infty}f(x)\,dx$ have the same convergence behavior.

When does a $p$-series converge?

The $p$-series $\sum_{n=1}^{\infty}\frac{1}{n^p}$ converges when $p>1$ and diverges when $p\le1$.

What are the conditions for the alternating series test?

The alternating series test requires eventually decreasing magnitudes $b_n$ and $b_n\to0$ for $\sum(-1)^{n-1}b_n$ to converge.

How can the error of an alternating-series approximation be bounded?

For an alternating series satisfying the test, the remainder after $N$ terms satisfies $|R_N|\le b_{N+1}$.

What are the outcomes of the ratio test?

For $L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$, the ratio test gives absolute convergence if $L<1$, divergence if $L>1$, and no conclusion if $L=1$.

What are the outcomes of the root test?

For $L=\lim_{n\to\infty}\sqrt[n]{|a_n|}$, the root test gives absolute convergence if $L<1$, divergence if $L>1$, and no conclusion if $L=1$.