# Thread: Series Help (I don't see how this adds up is all)

1. ## Series Help (I don't see how this adds up is all)

I'm trying to figure out how my teacher solved this infinite series, the problem is:

The sum as k = 1 goes to infinity of (1/3)^k,
so the Sequence of terms is: an = (1/3)^n,
and the Sequence of partial sums is S1 = 1/3, S2 = (1/3 + 1/9), S3 = etc.

Then my teacher goes to say that: Lim as n -> Infinity of Sn = 1/2 , and I have no idea how he got that, can anyone help me figure this out?

2. Originally Posted by vandorin
I'm trying to figure out how my teacher solved this infinite series, the problem is:

The sum as k = 1 goes to infinity of (1/3)^k,
so the Sequence of terms is: an = (1/3)^n,
and the Sequence of partial sums is S1 = 1/3, S2 = (1/3 + 1/9), S3 = etc.

Then my teacher goes to say that: Lim as n -> Infinity of Sn = 1/2 , and I have no idea how he got that, can anyone help me figure this out?

The sum of a FINITE geometric series is $a+aq+aq^2+\ldots +aq^{n-1}=\sum^{n-1}_{k=1}aq^k=a\frac{q^n-1}{q-1}$ .

Now, if $|q|<1$ then $q^n\xrightarrow [n\to\infty]{}0$ , so passing to the limit in the first sum we get:

$\sum^\infty_{k=0}aq^k=\lim_{n\to\infty}a\frac{q^n-1}{q-1}=-\frac{a}{q-1}=\frac{a}{1-q}$ .

Now check again your exercise, ony taken into account that your infinite sum begins with $k=1$ and NOT with $k=0$ .

Tonio

3. Originally Posted by tonio
The sum of a FINITE geometric series is $a+aq+aq^2+\ldots +aq^{n-1}=\sum^{n-1}_{k=1}aq^k=a\frac{q^n-1}{q-1}$ .

Now, if $|q|<1$ then $q^n\xrightarrow [n\to\infty]{}0$ , so passing to the limit in the first sum we get:

$\sum^\infty_{k=0}aq^k=\lim_{n\to\infty}a\frac{q^n-1}{q-1}=-\frac{a}{q-1}=\frac{a}{1-q}$ .

Now check again your exercise, ony taken into account that your infinite sum begins with $k=1$ and NOT with $k=0$ .

Tonio
Now it makes sense! Thank you!