# Is the series indicated absolutely convergent, conditionally convergent, or divergent? #rarr\4-1+1/4-1/16+1/64...#

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Use the appropriate test.

I get the feeling this is an Alternating Series, but I'm not sure if it's #(-1)^n# or #(-1)^(n+1)# or #(-1)^(n-1)# ??

The exponents seem to be decreasing, i.e. #a^1# , #a^0# , #a^-1# , #a^-2# ...

Use the appropriate test.

I get the feeling this is an Alternating Series, but I'm not sure if it's

The exponents seem to be decreasing, i.e.

The given series is a geometric series with the common ratio ( r = \frac{1}{4} ). It is conditionally convergent because its terms alternate in sign and the absolute value of its terms approaches zero as ( n ) approaches infinity.

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The geometric series converges absolutely, with

This series is definitely an alternating series; however, it also looks geometric.

If we can determine the common ratio shared by all terms, the series will be in the form

We'll need to find the summation using the above format.

We can write the series as

Now, let's determine if it converges absolutely.

Strip out the alternating negative term:

Take the absolute value, causing the alternating negative term to vanish:

Thus,

The series converges absolutely, with

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It converges absolutely.

Use the test for absolute convergence. If we take the absolute value of the terms we get the series

Hopefully this helps!

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The series is conditionally convergent.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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