Step By Step Calculus » 17.2 - Integral and Comparison Tests

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Synopsis
There exists a number of other specialized tests to determine whether a series is convergent or divergent. Here are three test mechanisms for nonnegative series.
Integral Test: Suppose \displaystyle \sum_{i=1}^{\infty}a_i\displaystyle \sum_{i=1}^{\infty}a_i is a nonnegative series and f:[0,\infty)\to[0,\infty)f:[0,\infty)\to[0,\infty) is monotonically decreasing with antiderivative FF on [0,\infty)[0,\infty) and satisfies f(i)=a_if(i)=a_i. Then,
\displaystyle s_n=\sum_{i=1}^na_i\in\left[F(n+1)-F(1), F(n)-F(0)\right]
\displaystyle s_n=\sum_{i=1}^na_i\in\left[F(n+1)-F(1), F(n)-F(0)\right]
and \displaystyle \sum_{i=1}^{\infty}a_i\displaystyle \sum_{i=1}^{\infty}a_i converges if and only if \displaystyle \lim_{t\to\infty} F(t) <\infty\displaystyle \lim_{t\to\infty} F(t) <\infty.
Comparison Test: For any N\in\mathbb{N}N\in\mathbb{N},
  • If 0\leq a_i\leq b_i\;0\leq a_i\leq b_i\;\forall i\geq N\forall i\geq N, then \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i converges if \displaystyle \sum_{i=1}^{\infty} b_i\displaystyle \sum_{i=1}^{\infty} b_i converges.
  • If 0\leq b_i\leq a_i\;0\leq b_i\leq a_i\;\forall i\geq N\forall i\geq N, then \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i diverges if \displaystyle \sum_{i=1}^{\infty} b_i\displaystyle \sum_{i=1}^{\infty} b_i diverges.
Limit Comparison Test: Consider two nonnegative series \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i and \displaystyle \sum_{i=1}^{\infty} b_i\displaystyle \sum_{i=1}^{\infty} b_i.
  • If \displaystyle \lim_{n\rightarrow\infty} \frac{a_n}{b_n} = c\displaystyle \lim_{n\rightarrow\infty} \frac{a_n}{b_n} = c for c\in[0,\infty)c\in[0,\infty) and \displaystyle \sum_{i=1}^{\infty} b_i\displaystyle \sum_{i=1}^{\infty} b_i converges, then \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i converges.
  • If \displaystyle \lim_{n\rightarrow\infty} \frac{a_n}{b_n} = c\displaystyle \lim_{n\rightarrow\infty} \frac{a_n}{b_n} = c for c\in(0,\infty)\cup\{\infty\}c\in(0,\infty)\cup\{\infty\} and \displaystyle \sum_{i=1}^{\infty} b_i\displaystyle \sum_{i=1}^{\infty} b_i diverges, then \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i diverges.
\mathbf{p}\mathbf{p}-Series Rule: A pp-series has the form \displaystyle \sum_{i=1}^{\infty}\frac{1}{i^p}\displaystyle \sum_{i=1}^{\infty}\frac{1}{i^p} and is convergent when p>1p>1 and divergent when p\leq 1p\leq 1.