Infinite Series Convergence – Calculus Tutorials

Convergence Tests for Infinite Series – HMC Calculus TutorialIn this tutorial, we review some of the most common tests for theconvergence of an infinite series$$\sum_{k=0}^{\infty} a_k = a_0 + a_1 + a_2 + \cdots$$The proofs or these tests are interesting, so we urge you to look themup in your calculus text.Let\begin{eqnarray*}s_0 & = & a_0 \\s_1 & = & a_1 \\& \vdots & \\s_n & = & \sum_{k=0}^{n} a_k \\& \vdots & \end{eqnarray*}If the sequence $\{ s_n \}$ of partial sums converges to a limit$L$, then the series is said to converge to the sum $L$and we write$\qquad$$$\sum_{k=0}^{\infty}a_k = L.$$$\qquad\qquad$For $j \ge 0$, $\sum\limits^{\infty}_{k=0} a_k$ converges if and onlyif $\sum\limits_{k=j}^{\infty} a_k$ converges, so in discussingconvergence we often just write $\sum a_k$.ExampleConsider the geometric series$$\sum_{k=0}^{\infty} x^k.$$The $n^{th}$ partial sum is$$s_n = 1 + x + x^{2} + \cdots + x^{n}.$$Multiplying both sides by $x$,$$xs_n = x + x^{2} + x^{3} + \cdots + x^{n+1}.$$S...