Question

How do i find the sum??

http://i.imgur.com/tBwQ0Fi.png

I know to represent it as a geometric series but I am unsure what to do with the ^n-1

Answer 1

We can use geometric series test, so first lets find the partial sum for \[\sum_{n=1}^{\infty} \frac{ (-4)^{n-1} }{ 5^n }\] so we let n = 1

Answer 2

\[a_1 = \frac{ (-4)^{1-1} }{ 5^1 } = \frac{ (-4)^0 }{ 5 } = \frac{ 1 }{ 5 }\] and we know if |r|<1 it is then convergent to find r we can use \[a_1r=a_2 \implies \frac{ 1 }{ 5 }r=-\frac{ 4 }{ 25 } \implies r = -\frac{ 20 }{ 25 } = -\frac{ 4 }{ 5 }\]

\[\frac{ a_1 }{ 1-r } = \frac{ 1/5 }{ 1+4/5 } = \frac{ 1 }{ 9 }\] therefore it's convergent

Answer 3

Your sum is 1/9

Answer 4

Oh i see we manipulated a2/a1=r

Answer 5

Yup

Answer 6

Why did we use n=1?

Answer 7

That gives us our first term a_1

Answer 8

Remember r is the common ratio here

Answer 9

Oh right n=1 in the sum so its our first term, i keep on thinking it starts at zero.

Thanks

Answer 10

Np :)