# It is important to distinguish between sum_{n=1}^{infty} n^{b} quad text { and } quad sum_{n=1}^{infty} b^{n} What name is given to the first series? To the second? For what values of b does the first series converge? For what values of b does the second series converge?

It is important to distinguish between

What name is given to the first series? To the second? For what values of b does the first series converge? For what values of b does the second series converge?
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Given:
The series
As,
$\sum _{n=1}^{\mathrm{\infty }}{n}^{b}={1}^{b}+{2}^{b}+{3}^{b}+{4}^{b}+...$
Thus it is an arithmetic increasing power series.
Similarly,
$\sum _{n=1}^{\mathrm{\infty }}{b}^{n}=b+{b}^{2}+{b}^{3}+{b}^{4}+...$
Here first term is b and common ratio is also b.
Thus it is a geometric power series.
Ratio test:
Consider the series: $\sum _{n=1}^{\mathrm{\infty }}{a}^{n}$
Let $L=\underset{n\to \mathrm{\infty }}{lim}|\frac{{a}_{n+1}}{{a}_{n}}|$
If L< 1, then series is convergent.
If L>1, then series is divergent and
If L=1, then test is inconclusive.
For $\sum _{n=1}^{\mathrm{\infty }}{n}^{b}$ the nth term ${a}_{n}={n}^{b}$
Thus $L=\underset{n\to \mathrm{\infty }}{lim}|\frac{\left(n+1{\right)}^{b}}{{n}^{b}}|$
$=\underset{n\to \mathrm{\infty }}{lim}|\left(\frac{n+1}{n}{\right)}^{b}|$
$=\underset{n\to \mathrm{\infty }}{lim}|\left(1+\frac{1}{n}{\right)}^{b}|$
$=\left(1+0{\right)}^{b}$
$=1$
As L=1, test is inconclusive.
Series is convergent if $L<1$.
But for any value of b it will never be less than 1.
Therefore, for any value of b, the series $\sum _{n=1}^{\mathrm{\infty }}{n}^{b}$ is not convergent.
For $\sum _{n=1}^{\mathrm{\infty }}{b}^{n}$ the nth term of series is ${a}_{n}={b}^{n}$
Thus
$L=\underset{n\to \mathrm{\infty }}{lim}|\frac{{b}^{n+1}}{{b}^{n}}|$
$=\underset{n\to \mathrm{\infty }}{lim}|b|$
$=b$
Series is convergent if $L<1$.
Thus $b<1$.
Therefore, for $b<1$, the series $\sum _{n=1}^{\mathrm{\infty }}{b}^{n}$ is convergent.

Jeffrey Jordon