# Find the values of x for which the series converges, find the sum of those values.

• Mar 20th 2013, 05:25 PM
Coop
Find the values of x for which the series converges, find the sum of those values.
Hi,

Here's the question,

"Find the values of x for which the series converges. Find the sum of the series for those values of x.
summation(n=0,inf) [(x+3)^n]/2^n"

I wasn't really sure how to approach it. I realized that it can become a convergent geometric series if |x+3| < 2, because that would make the absolute value of the r value less than 1.

Taking this into account, I am proposing that -5< x < -1

I am pretty sure am I right up to this point after looking around online, but I am confused about the second part. I thought maybe you could find the infinite sum when x is -5 and when x is -1 and then subtract the first value from the second value. Is this the correct way to go about it? I got 8/21 as an answer.

Thanks
• Mar 20th 2013, 05:47 PM
HallsofIvy
Re: Find the values of x for which the series converges, find the sum of those values
Quote:

Originally Posted by Coop
Hi,

Here's the question,

"Find the values of x for which the series converges. Find the sum of the series for those values of x.
summation(n=0,inf) [(x+3)^n]/2^n"

So $\displaystyle \sum_{n=0}^\infty \left(\frac{x+ 3}{2}\right)^n$

Quote:

I wasn't really sure how to approach it. I realized that it can become a convergent geometric series of |x+3| < 2, because that would make the absolute value of the r value = 1.
Yes, that is a geometric sequence with "common ratio" $\displaystyle \frac{x+ 3}{2}$ which must be between -1 and 1 in order that it converge absolutely.

Quote:

Taking this into account, I am proposing that -5< x < -1
Okay, $\displaystyle -1< \frac{x+ 3}{2}< 1$ so that -2< x+ 3< 2, -5< x< -1.

Quote:

I am pretty sure am I right up to this point after looking around online, but I am confused about the second part. I thought maybe you could find the infinite sum when x is -5 and when x is -1 and then subtract the first value from the second value. Is this the correct way to go about it? I got 8/21 as an answer.

Thanks
I am afraid you have completely misunderstood this part of the question. It is asking for the sum as a function of x, not a difference between two values. You have correctly identified it as a geometric sequence, $\displaystyle \sum_{n=0}^n ar^n$, with common ratio $\displaystyle r= \frac{x+3}{2}$ and first term a= 1. Now, recall that the sum of such a geometric series is $\displaystyle \frac{a}{1- r}$.
• Mar 20th 2013, 06:33 PM
Coop
Re: Find the values of x for which the series converges, find the sum of those values
Quote:

Originally Posted by HallsofIvy
I am afraid you have completely misunderstood this part of the question. It is asking for the sum as a function of x, not a difference between two values. You have correctly identified it as a geometric sequence, $\displaystyle \sum_{n=0}^n ar^n$, with common ratio $\displaystyle r= \frac{x+3}{2}$ and first term a= 1. Now, recall that the sum of such a geometric series is $\displaystyle \frac{a}{1- r}$.

Isn't the "a" value whatever you have when n=1, so wouldn't "a" be (x+3)/2 and since the whole sequence is in parenthesis, that would be the "r" value as well?
• Mar 20th 2013, 07:19 PM
Prove It
Re: Find the values of x for which the series converges, find the sum of those values
No, the a value is the starting value. Your first term has n = 0, not n = 1.
• Mar 20th 2013, 07:25 PM
Coop
Re: Find the values of x for which the series converges, find the sum of those values
Quote:

Originally Posted by Prove It
No, the a value is the starting value. Your first term has n = 0, not n = 1.

Oh of course, I forgot about the 0, thanks :)
• Mar 21st 2013, 07:18 AM
Coop
Re: Find the values of x for which the series converges, find the sum of those values
Quote:

Originally Posted by Prove It
No, the a value is the starting value. Your first term has n = 0, not n = 1.

So the sum = 1/(1-((x+3)/2)), that's the final answer?
• Mar 21st 2013, 08:42 AM
HallsofIvy
Re: Find the values of x for which the series converges, find the sum of those values
Well, do the algebra now! What does that reduce to?
• Mar 21st 2013, 08:45 AM
Coop
Re: Find the values of x for which the series converges, find the sum of those values
Quote:

Originally Posted by HallsofIvy
Well, do the algebra now! What does that reduce to?

2/(x-1), right?

Thanks for the help :)
• Mar 21st 2013, 02:30 PM
Prove It
Re: Find the values of x for which the series converges, find the sum of those values
Try again...
• Mar 21st 2013, 02:37 PM
Coop
Re: Find the values of x for which the series converges, find the sum of those values
Quote:

Originally Posted by Prove It
Try again...

Yeah I forgot about the parenthesis :/ Anyway, thanks.