Solving Series: Calculate ##\sum\frac{4^{n+1}}{5^n}##

[goraemon]

Homework Statement

Calculate $\sum\frac{4^{n+1}}{5^n}$ (where n begins at 0 and approaches infinity).

Homework Equations

The Attempt at a Solution

I could easily solve this if the numerator were just $4^n$ instead of $4^{n+1}$, because then it would be a geometric series with ratio of $\frac{4}{5}$. But I'm not sure how to approach this one. Any help would be appreciated.

 

[UltrafastPED]

What happens if you pull a 4 from every term - starting from n=1?

 

[goraemon]

What happens if you pull a 4 from every term - starting from n=1?

Oh...so the problem becomes...
$4*\sum(\frac{4}{5})^n=4*\frac{1}{1-\frac{4}{5}}=4*5=20$
Is that right? Thanks!

 

[Ray Vickson]

Oh...so the problem becomes...
$4*\sum(\frac{4}{5})^n=4*\frac{1}{1-\frac{4}{5}}=4*5=20$
Is that right? Thanks!

Well, do YOU think it is right?

 

[goraemon]

Well, do YOU think it is right?

...yes? But then again, I wouldn't be here if I were always right just because I think I am.

 

[Ray Vickson]

...yes? But then again, I wouldn't be here if I were always right just because I think I am.

What I am suggesting is that you develop some confidence in your own work. If you do things carefully, without making a mistake at any step and without violating any "rules" you are 100% guaranteed to have the correct answer. If the problem is simple enough it should be easy for you to check your own work (and that is a something you should always do, anyway); of course, for a complicated and lengthy problem the situation is different, and developing self-confidence is harder for those cases. Remember: you need to be able to do these things in an exam.

BTW: yes, it is correct.