Derivative of cosh x: Clarifying Confusion

[Michael_Light]

Hi... I read this from my lecture note, but i couldn't understand:

i) the red part, shouldn't it be k =0 instead of k=1?

ii) the third line to the fourth line... i.e.

I have no idea how to change from the third line to the fourth line..

Can anyone enlighten me? Thanks a lot..

 

[HallsofIvy]

It doesn't matter. Since the terms being summed have a factor of "k", the k= 0 term has value 0.

In general, for power series, f(x)= \sum_{k=0}^\infty a_kx^k, the derivative can be written as either f'(x)= \sum_{k=0}^\infty ka_kx^{k- 1} or as f'(x)= \sum_{k=1}^\infty ka_kx^{k- 1} because the k= 0 term is 0.

By making the change of index, n= k- 1, so that k= n+1, the second can be written f'(x)= \sum_{n=0} (n+1)a_{n+1}x^n.

 

[THSMathWhiz]

The first sum can be rewritten as

\displaystyle \sum_{k=1}^\infty (2k)\frac{x^{2k-1}}{(2k)!}=\sum_{k=1}^\infty \frac{x^{2k-1}}{(2k-1)!}​

since (2k)!=(2k)(2k-1)!. To change the base index, replace k by k+1. You then get the sum

\displaystyle \sum_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}.​

 

[mathman]

View attachment 54593

Hi... I read this from my lecture note, but i couldn't understand:

i) the red part, shouldn't it be k =0 instead of k=1?

ii) the third line to the fourth line... i.e.

View attachment 54595

I have no idea how to change from the third line to the fourth line..

Can anyone enlighten me? Thanks a lot..

This seems to be a very roundabout way of getting this result.

cosh(x) = (e^x + e^-x)/2
The derivative = (e^x - e^-x)/2 =sinh(x).

 

[HallsofIvy]

IF cosh(x) has been defined as (e^x+ e^{-x})/2, yes, that is more direct. However, if cosh(x) has been defined as \sum_{i=0}^\infty x^{2i}/(2i)!, as is quite possible and apparently as done in the first post, the argument given is more direct.