Solving Nth Term of a Series Homework

[Aimee79]

Homework Statement

Not really sure where tou go with this one.

Homework Equations

If the nth partial sum of a partial series is given by,

Sn= $$\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}$$ = $$\sum_{k=1}^{n}$$$$\frac{1}{n+k}$$

a) write the associated series
b) test for convergence
c) if possible, determine its limit

The Attempt at a Solution

Here is what I have come up with:
$$s_1$$=1/2
$$s_2$$=1/2+1/3+1/4
$$s_3$$=1/2+1/3+1/4+1/5+1/6

$$a_n = \left\{ \begin{array}{c} \frac{1}{2} \text{ for }n=1 \\ \frac{1}{2n-1}+\frac{1}{2n} \text{ for }n\geq 2 \end{array} \right.$$

I don't know what to do next. What do I do with the 1/2?

I am pretty sure I can handle b and c, I just need help with a.

Thanks in advance!

 

[Gib Z]

You seem to have the general term. Really, you just now just say that associated series is sum of a_n, but if you want to put it "nicely", then putting in n=1 in the expression you have for n>2, its equal to 1.5, which is 1 too big. So you can use the same generating rule for all n, as long as you subtract 1 from the series.

 

[tiny-tim]

"associated series"

a) write the associated series
…
I am pretty sure I can handle b and c, I just need help with a.

Hi Aimee79!

The associated series just means the series {a_n} such that a₁ + … + a_n = S_n.

Hint: subtract something.