# How to prove that the series is converge?

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#sum (cos (1/k))/(9k^2)#

Converges by the Direct Comparison Test.

We can use the Direct Comparison Test, so far as we have

Then, we can define a new sequence

Well,

Then, since the larger series converges, so must the smaller series.

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It converges by the direct comparison test (see below for details).

graph{cos(1/x) [-10, 10, -5, 5]}

As you can see, the maximum value this will achieve will be 1. Since we're just trying to prove convergence here, let's set the numerator to 1, leaving:

Now, this becomes a very simple direct comparison test problem. Recall what the direct comparison test does:

Hope that helped :)

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To prove that a series converges, you can use various convergence tests such as the comparison test, ratio test, root test, integral test, or alternating series test. Each test has its conditions and method of application. You need to determine which test is applicable to the given series based on its characteristics. Then, follow the steps of the chosen test to demonstrate convergence. If the conditions of the test are met and the series satisfies the convergence criterion, you can conclude that the series converges.

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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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