William Cleghorn

2022-01-06

What is the appropriate way to simplify such an expression. i am unsure of how to use the series i know to apply to this situation

$\sum _{L=0}^{M}{s}^{L}{L}^{2}$

aquariump9

Beginner2022-01-07Added 40 answers

Write : ${L}^{2}=L(L-1)+L$ and use derivative. For $L\ge 2$ :

$L}^{2}{s}^{L}={s}^{2}L(L-1){s}^{L-2}+sL{s}^{L-1}={s}^{2}\left({s}^{L}\right){}^{\u2033}+s{\left({s}^{L}\right)}^{\prime$

We get:

$\sum _{L=0}^{M}{L}^{2}{s}^{L}={0}^{2}+{1}^{2}s+{s}^{2}\left(\sum _{L=2}^{M}{s}^{L}\right){}^{\u2033}+s{\left(\sum _{L=2}^{M}{s}^{L}\right)}^{\prime}$

We get:

Neunassauk8

Beginner2022-01-08Added 30 answers

Try to make the inner expression look like a derivative:

$\sum _{L=0}^{M}\left(L{s}^{L-1}\right)sL=s\sum _{L=0}^{M}\left({d}_{s}{s}^{L}\right)L$

$=s{d}_{s}\sum _{L=0}^{M}{s}^{L}L$

$=s{d}_{s}\sum _{L=0}^{M}\left(L{s}^{L-1}\right)s$

$=s{d}_{s}\left(s\sum _{L=0}^{M}\left(L{s}^{L-1}\right)\right)$

$=s{d}_{s}\left(s\sum _{L=0}^{M}{d}_{s}{s}^{L}\right)$

$=s{d}_{s}\left(s{d}_{s}\sum _{L=0}^{M}{s}^{L}\right)$

$=s{d}_{s}\left(s{d}_{s}\frac{{s}^{M+1}-1}{s-1}\right)$

Now just take it from here, simplifying from the inside out.

Now just take it from here, simplifying from the inside out.

karton

Expert2022-01-11Added 613 answers

Rewrite

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