Example 13.5.1 Does $\ds\sum_{n=2}^\infty {1\over n^2\ln n}$ converge?

The obvious first approach, based on what we know, is the integral test. Unfortunately, we can't compute the required antiderivative. But looking at the series, it would appear that it must converge, because the terms we are adding are smaller than the terms of a $p$-series, that is, $${1\over n^2\ln n}< {1\over n^2},$$ when $n\ge3$. Since adding up the terms $\ds 1/n^2$ doesn't get "too big'', the new series "should'' also converge. Let's make this more precise.

The series $\ds\sum_{n=2}^\infty {1\over n^2\ln n}$ converges if and only if $\ds\sum_{n=3}^\infty {1\over n^2\ln n}$ converges—all we've done is dropped the initial term. We know that $\ds\sum_{n=3}^\infty {1\over n^2}$ converges. Looking at two typical partial sums: $$ s_n={1\over 3^2\ln 3}+{1\over 4^2\ln 4}+{1\over 5^2\ln 5}+\cdots+ {1\over n^2\ln n} < {1\over 3^2}+{1\over 4^2}+ {1\over 5^2}+\cdots+{1\over n^2}=t_n. $$ Since the $p$-series converges, say to $L$, and since the terms are positive, $\ds t_n< L$. Since the terms of the new series are positive, the $\ds s_n$ form an increasing sequence and $\ds s_n< t_n< L$ for all $n$. Hence the sequence $\ds \{s_n\}$ is bounded and so converges. $\square$

Example 13.5.2 Does $\ds\sum_{n=1}^\infty {|\sin n|\over n^2}$ converge?

We can't apply the integral test here, because the terms of this series are not decreasing. Just as in the previous example, however, $$ {|\sin n|\over n^2}\le {1\over n^2},$$ because $|\sin n|\le 1$. Once again the partial sums are non-decreasing and bounded above by $\ds \sum 1/n^2=L$, so the new series converges. $\square$

Example 13.5.3 Does $\ds\sum_{n=2}^\infty {1\over\sqrt{n^2-3}}$ converge?

We observe that the $-3$ should have little effect compared to the $\ds n^2$ inside the square root, and therefore guess that the terms are enough like $\ds 1/\sqrt{n^2}=1/n$ that the series should diverge. We attempt to show this by comparison to the harmonic series. We note that $${1\over\sqrt{n^2-3}} > {1\over\sqrt{n^2}} = {1\over n},$$ so that $$ s_n={1\over\sqrt{2^2-3}}+{1\over\sqrt{3^2-3}}+\cdots+ {1\over\sqrt{n^2-3}} > {1\over 2} + {1\over3}+\cdots+{1\over n}=t_n, $$ where $\ds t_n$ is 1 less than the corresponding partial sum of the harmonic series (because we start at $n=2$ instead of $n=1$). Since $\ds\lim_{n\to\infty}t_n=\infty$, $\ds\lim_{n\to\infty}s_n=\infty$ as well. $\square$

Example 13.5.4 Does $\ds\sum_{n=1}^\infty {1\over\sqrt{n^2+3}}$ converge?

Just as in the last example, we guess that this is very much like the harmonic series and so diverges. Unfortunately, $${1\over\sqrt{n^2+3}} < {1\over n},$$ so we can't compare the series directly to the harmonic series. A little thought leads us to $${1\over\sqrt{n^2+3}} > {1\over\sqrt{n^2+3n^2}} = {1\over2n},$$ so if $\sum 1/(2n)$ diverges then the given series diverges. But since $\sum 1/(2n)=(1/2)\sum 1/n$, theorem 13.2.2 implies that it does indeed diverge. $\square$