Determining whether a sequence of partial sums is convergent or divergent

Given the following sequence of partial sums:

My attempt:

If I can find the series for this sequence of partial sums, then I can test the series to see if it's convergent or divergent.

Thus:

Since :

But this is as far as I got.

I would like to express this with a factor so that I can use the alternating series test, but I'm not sure how exactly.

Would this be correct?

Re: Determining whether a sequence of partial sums is convergent or divergent

Quote:

Originally Posted by

**MathIsOhSoHard** Given the following sequence of partial sums:

If the real question is contained in the title of this thread, then you have wasted a lot effort.

Because the partial sums are given as .

You know so the series converges to zero.

If the question asks something else, then the title is misleading.

Re: Determining whether a sequence of partial sums is convergent or divergent

This is the question in its entirety:

For a series the sequence of partial sums is . Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

Hint: Find the series n'th term and express it as an alternating series.

(It's supposed to be an infinity sign above the sum, but for some reason LATEX doesn't want to display it)

Re: Determining whether a sequence of partial sums is convergent or divergent

Quote:

Originally Posted by

**MathIsOhSoHard** This is the question in its entirety:

For a series

the sequence of partial sums is

. Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

Hint: Find the series n'th term

and express it as an alternating series.

Well the only difficult part is "Determine whether the series is absolutely convergent,"

For that reason you may want to find .

.

[tex]\sum\limits_{n = 0}^\infty {ar^n } [/tex] gives

Re: Determining whether a sequence of partial sums is convergent or divergent

Quote:

Originally Posted by

**Plato** Well the only difficult part is "Determine whether the series is absolutely convergent,"

For that reason you may want to find

.

.

[tex]\sum\limits_{n = 0}^\infty {ar^n } [/tex] gives

Are you sure can't be written as

Re: Determining whether a sequence of partial sums is convergent or divergent

Quote:

Originally Posted by

**MathIsOhSoHard** Are you sure

can't be written as

**YES**

It is

You made a *sign error*.

Re: Determining whether a sequence of partial sums is convergent or divergent

Quote:

Originally Posted by

**Plato** **YES**
It is

You made a

*sign error*.

How do you rewrite into ? :)

Re: Determining whether a sequence of partial sums is convergent or divergent

Quote:

Originally Posted by

**mathisohsohard** how do you rewrite

into

? :)

You see

Re: Determining whether a sequence of partial sums is convergent or divergent

Great! Now I get it :)

So by using the alternating series test, it can be shown that the series is convergent.

And to test for absolute convergence, using the comparison test:

Since is divergent, then the absolute of is divergent too, thus it is not absolutely convergent.

So conclusion would be that it's conditionally convergent?

Would the absolute value of the fraction be or ? I wasn't quite sure about that.

Re: Determining whether a sequence of partial sums is convergent or divergent

Quote:

Originally Posted by

**MathIsOhSoHard** Great! Now I get it :)

So by using the alternating series test, it can be shown that the series is convergent.

And to test for absolute convergence, using the comparison test:

Since

is divergent, then the absolute of

is divergent too, thus it is not absolutely convergent.

So conclusion would be that it's conditionally convergent?

Would the absolute value of the fraction be

or

? I wasn't quite sure about that.

Very good.