Lubna Boyce

2022-02-28

I am given the series

$\sum _{n=1}^{\mathrm{\infty}}\frac{{n}^{n-1}}{{(2{n}^{2}+n+1)}^{(n+\frac{1}{2})}}$

shotokan0758s

Beginner2022-03-01Added 8 answers

The ratio test works fine

$a}_{n}=\frac{{n}^{n-1}}{{(2{n}^{2}+n+1)}^{(n+\frac{1}{2})}$

Take loragithms

$\mathrm{log}\left({a}_{n}\right)=(n-1)\mathrm{log}\left(n\right)-(n+\frac{1}{2})\mathrm{log}(2{n}^{2}+n+1)$

Using Taylor for large values of n.

$\mathrm{log}\left({a}_{n}\right)=n\mathrm{log}\left(\frac{1}{2n}\right)+\frac{1}{2}(\mathrm{log}\left(\frac{1}{2{n}^{4}}\right)-1)-\frac{5}{8n}+O\left(\frac{1}{{n}^{2}}\right)$

Apply it twice and continue with Taylor series

$\mathrm{log}\left({a}_{n+1}\right)-\mathrm{log}\left({a}_{n}\right)=(\mathrm{log}\left(\frac{1}{2n}\right)-1)-\frac{5}{2n}$

$\frac{{a}_{n+1}}{{a}_{n}}={e}^{\mathrm{log}\left({a}_{n+1}\right)-\mathrm{log}\left({a}_{n}\right)}=\frac{1}{2en}(1-\frac{5}{2en}+O\left(\frac{1}{{n}^{2}}\right))$

Now, using the expansion of$\mathrm{log}\left({a}_{n}\right)$ to $O\left(\frac{1}{n}\right)$ , we then have your formula

$a}_{n}\sim \frac{1}{\sqrt{2e}}\frac{1}{{2}^{n}{n}^{n+2}$

which is an overestimate of the true$a}_{n$

Take loragithms

Using Taylor for large values of n.

Apply it twice and continue with Taylor series

Now, using the expansion of

which is an overestimate of the true

What is the derivative of the work function?

How to use implicit differentiation to find $\frac{dy}{dx}$ given $3{x}^{2}+3{y}^{2}=2$?

How to differentiate $y=\mathrm{log}{x}^{2}$?

The solution of a differential equation y′′+3y′+2y=0 is of the form

A) ${c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

B) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{3x}$

C) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{-2x}$

D) ${c}_{1}{e}^{-2x}+{c}_{2}{2}^{-x}$How to find instantaneous velocity from a position vs. time graph?

How to implicitly differentiate $\sqrt{xy}=x-2y$?

What is 2xy differentiated implicitly?

How to find the sum of the infinite geometric series given $1+\frac{2}{3}+\frac{4}{9}+...$?

Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?

A. 4.2

B. 4.4

C. 4.7

D. 5.1What is the derivative of $\frac{x+1}{y}$?

How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

How to find the volume of a cone using an integral?

What is the surface area of the solid created by revolving $f\left(x\right)={e}^{2-x},x\in [1,2]$ around the x axis?

How to differentiate ${x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}=4$?

The differential coefficient of $\mathrm{sec}\left({\mathrm{tan}}^{-1}\left(x\right)\right)$.