Series help

[EgamiRorrim]

There are two problems that I need help with that deal with series. They are both summation problems, but I do not know how to put them in sigma notation on this forum.

1) sum(1/4^i, i = 0 .. infinity)

2) sum(i/4^i, i = 0 .. infinity)

 

[pka]

There are two problems that I need help with that deal with series. They are both summation problems, but I do not know how to put them in sigma notation on this forum.
1) sum(1/4^i, i = 0 .. infinity)
2) sum(i/4^i, i = 0 .. infinity)

[tex]S=\sum\limits_{n = 0}^\infty {r^n }=\frac{1}{1-r} [/tex] gives \(\displaystyle S=\sum\limits_{n = 0}^\infty {r^n }=\frac{1}{1-r} \).

So \(\displaystyle S'=\sum\limits_{n = 0}^\infty {nr^{n-1} }=\frac{1}{(1-r)^2} \).

So \(\displaystyle rS’=\sum\limits_{n = 0}^\infty {nr^{n} }=\frac{r}{(1-r)^2} \).

 

[lookagain]

[tex]S=\sum\limits_{n = 0}^\infty {r^n }=\frac{1}{1-r} [/tex] gives \(\displaystyle S=\sum\limits_{n = 0}^\infty {r^n }=\frac{1}{1-r} \).

So \(\displaystyle S'=\sum\limits_{n = 0}^\infty {nr^{n-1} }=\frac{1}{(1-r)^2} \).

So \(\displaystyle rS'=\sum\limits_{n = 0}^\infty {nr^{n}} }=\frac{r}{(1-r)^2} \).

I made an amendment above in the quote box.

The prime mark on the last "S" did not show in the post,
because it was an apostrophe instead of the different,
but similar, character used in the line just above it.