Convergent series

A series converges when its sequence of partial sums converges to a finite limit.

If the partial sums of $\sum b_n$ are bounded and $a_n$ decreases to zero, then $\sum a_n{b}_n$ converges. The uniform version holds for functions $b_n(x)$ when their partial sums have one bound $M$ independent of $x$. Indeed partial sums starting at index $N$ have absolute value at most $2M$, and summation by parts bounds every weighted tail from $N$ to $K$ by $2M{a}_N$. The uniformly Cauchy sequence criterion proves uniform convergence. This justifies integrating a conditionally convergent Fourier series on intervals staying away from its endpoint jump.

For nonnegative sequences with $0\le a_n\le b_n$, convergence of ${\sum }_n{b}_n$ implies convergence of ${\sum }_n{a}_n$, while divergence of ${\sum }_n{a}_n$ implies divergence of ${\sum }_n{b}_n$.

If $f:[1,\infty )\to [0,\infty )$ is decreasing, then ${\sum }_{n\ge 1}f(n)$ and ${\int }_1^{\infty }f(x)dx$ either both converge or both diverge. This follows by bounding the area under the graph between adjacent upper and lower rectangle sums.

A sequence $(a_n)$ is summable when the series ${\sum }_n{a}_n$ is convergent.

For positive terms, if the ratio of two sequences tends to a finite positive number, their series either both converge or both diverge.

For nonzero terms, ifthen $\sum a_n$ converges absolutely. If the ratio has a limit greater than one, the terms do not tend to zero and the series diverges.

For a positive sequence, if $a_{n+1}/a_n\to L\ge 0$, thenFor $L>0$ this follows by taking logarithms and applying Cesàro averaging; the case $L=0$ follows from an eventual geometric upper bound.

For $a_n=n!/n^n$,Thus $\sum n!/n^n$ converges and $n/(n!{)}^{1/n}\to e$.