Thread: Calc II: Finding a sequence from a partial sum equation

1. Calc II: Finding a sequence from a partial sum equation

I don't even know where to start for this question. If someone could help me with the first few steps, I would be very grateful!

If the nth partial sum of a series Sigma (infinite sum, n=1) is s(sub)n=3-n^(-n), find a(sub)n and the partial sum of a(sub)n

2. Re: Calc II: Finding a sequence from a partial sum equation

Start at the beginning.

What is a partial sum? What is meant by $s_n\,?$

3. Re: Calc II: Finding a sequence from a partial sum equation

A partial sum is taking all integers (to infinity) plugging them into an equation, and adding each term together. I started with a table, and tried to find some sort of pattern by subtraction between each term, but got nowhere fast.

4. Re: Calc II: Finding a sequence from a partial sum equation

Originally Posted by clarinetqueen92
A partial sum is taking all integers (to infinity) plugging them into an equation, and adding each term together. I started with a table, and tried to find some sort of pattern by subtraction between each term, but got nowhere fast.
Gee, I thought the partial sum was the result of adding the first n terms of the series

$s_n=a_1+a_2+a_3+\dots+a_{n-2}+a_{n-1}+a_n=\sum_{k=1}^n\,a_k$

or was I wrong about that?

If I'm right, then $a_1=s_1=3-(1^{-1})=2$
$a_2=s_2-s_1=3-(2^{-2})-2=3-\frac{1}{4}-2$

etc.

5. Re: Calc II: Finding a sequence from a partial sum equation

Ok, our variables are a bit different. On your sigma, my n=infinity, k=1, and a(sub)k=3-k2^(-k). The question asks me to find a(sub)k and the sum from k=1 to infinity of a(sub)k.

(Sorry, I don't know how to use online math notations at all.)

Edit: Wait a minute, I missed some of your post. Hold on.

Edit 2: Ok, so we agree on our definitions of partial sums, but this is an infinite sum, and I don't understand any more about the problem than what I started out with... Sorry if I'm being dense...

6. Re: Calc II: Finding a sequence from a partial sum equation

do you want to find $\sum_{k=1}^{\infty}(3-k2^{-k})$?

If yes then you should take $\lim_{n\rightarrow\infty}\sum_{k=1}^{n}(3-k2^{-k})$ $=\lim_{n\rightarrow\infty}2^{-n} (3 \times 2^n n+n-2^{n+1}+2)$

Can you make something out of it?

7. Re: Calc II: Finding a sequence from a partial sum equation

Originally Posted by clarinetqueen92
Ok, our variables are a bit different. On your sigma, my n=infinity, k=1, and a(sub)k=3-k2^(-k). ...
Is that a typo in your a_k ? Does that 2 belong there?

8. Re: Calc II: Finding a sequence from a partial sum equation

I found that the series $\sum_{k=1}^{\infty}(3-\frac{k}{2^k})$ never converges but $\sum_{k=1}^{\infty}\frac{k}{2^k}=2$.