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- Homework Statement
- Find coefficient which multiply ##\frac{1}{p}## in product of series. Coefficient is function of ##t##.

[tex]\frac{1}{2}\sum^{\infty}_{n=0}\frac{(-1)^n}{n+1}(\frac{1}{p^2})^{n+1}\,\sum^{\infty}_{k=0}\frac{p^kt^k}{k!}[/tex]

Namely product of two series will be new series that will contain the sum of terms of ##p^l## where ##l## are integers. I want to find only the term ##f(t)\frac{1}{p}## in the product.

- Relevant Equations
- I think that the sum of terms that multiply ##1/p## in product of two series should be ##\frac{1-\cos t}{t}##.

First series

[tex]\frac{1}{2}\sum^{\infty}_{n=0}\frac{(-1)^n}{n+1}(\frac{1}{p^2})^{n+1}= \frac{1}{2}(\frac{1}{p^2}-\frac{1}{2p^4}+\frac{1}{3p^6}-\frac{1}{4p^8}+...)[/tex]

whereas second one is

[tex]\sum^{\infty}_{k=0}\frac{p^kt^k}{k!}=1+pt+\frac{p^2t^2}{2!}+\frac{p^3t^3}{3!}+\frac{p^4t^4}{4!}+\frac{p^5t^5}{5!}+...[/tex].

I multiply two series and I want to get coefficient that multiply ##1/p## in the series that I get as a product. I will get this type of term if I multiply ##\frac{1}{p^2}## with ##p##, ##\frac{1}{p^4}## with ##p^3##... And I need to sum all these terms.

I want to see what is coefficient that multiplies ##\frac{1}{p}## in the expansion of product of two series. That coefficient is function of ##t##.

First term I get by multiplying ##\frac{1}{2p^2}## and ##pt## and it is ##\frac{t}{2}##.

Second term I get by multiplying ##-\frac{1}{4p^4}## and ##\frac{p^3t^3}{3!}## and it is ##-\frac{t^3}{4!}##

Third term will be ##\frac{t^5}{6!}##...

I think that the sum should be ##\frac{1-\cos t}{t}##. Is there some easy way to get it without writing down all the sums?

[tex]\frac{1}{2}\sum^{\infty}_{n=0}\frac{(-1)^n}{n+1}(\frac{1}{p^2})^{n+1}= \frac{1}{2}(\frac{1}{p^2}-\frac{1}{2p^4}+\frac{1}{3p^6}-\frac{1}{4p^8}+...)[/tex]

whereas second one is

[tex]\sum^{\infty}_{k=0}\frac{p^kt^k}{k!}=1+pt+\frac{p^2t^2}{2!}+\frac{p^3t^3}{3!}+\frac{p^4t^4}{4!}+\frac{p^5t^5}{5!}+...[/tex].

I multiply two series and I want to get coefficient that multiply ##1/p## in the series that I get as a product. I will get this type of term if I multiply ##\frac{1}{p^2}## with ##p##, ##\frac{1}{p^4}## with ##p^3##... And I need to sum all these terms.

I want to see what is coefficient that multiplies ##\frac{1}{p}## in the expansion of product of two series. That coefficient is function of ##t##.

First term I get by multiplying ##\frac{1}{2p^2}## and ##pt## and it is ##\frac{t}{2}##.

Second term I get by multiplying ##-\frac{1}{4p^4}## and ##\frac{p^3t^3}{3!}## and it is ##-\frac{t^3}{4!}##

Third term will be ##\frac{t^5}{6!}##...

I think that the sum should be ##\frac{1-\cos t}{t}##. Is there some easy way to get it without writing down all the sums?

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