Using the integral test, how do you show whether #sum (1/n^2)cos(1/n) # diverges or converges from n=1 to infinity?
The
Therefore:
is a sum of nonnegative terms, bounded above and therefore convergent.
Hence:
is convergent.
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If you insist on using the integral test, you need to find
The integral converges, so the series also converges.
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To determine whether the series ( \sum \frac{1}{n^2} \cos\left(\frac{1}{n}\right) ) converges or diverges, we can use the integral test.

We start by considering the function ( f(x) = \frac{1}{x^2} \cos\left(\frac{1}{x}\right) ).

We need to check if ( f(x) ) is continuous, positive, and decreasing for all ( x ) from 1 to infinity.

The function ( f(x) ) is continuous and positive for ( x \geq 1 ).

To check if ( f(x) ) is decreasing, we examine its derivative ( f'(x) ).

Calculating the derivative ( f'(x) ), we find that it is not straightforward due to the product rule and the chain rule. However, we can observe that the cosine function oscillates between 1 and 1 as ( \frac{1}{x} ) oscillates between 1 and 0, resulting in alternating signs for the terms of the series.

Since ( \frac{1}{x^2} ) decreases as ( x ) increases, and the oscillation of ( \cos\left(\frac{1}{x}\right) ) does not affect the decreasing behavior of ( f(x) ) significantly, we can conclude that ( f(x) ) is decreasing for ( x \geq 1 ).

Now, we proceed to evaluate the integral ( \int_{1}^{\infty} f(x) , dx ).

This integral represents the area under the curve of ( f(x) ) from 1 to infinity.

Integrating ( f(x) ) analytically may not be feasible due to the complexity of the cosine function within the integral.

However, we can estimate the integral numerically using computational methods or software.

If the integral ( \int_{1}^{\infty} f(x) , dx ) converges, then the series ( \sum \frac{1}{n^2} \cos\left(\frac{1}{n}\right) ) converges by the integral test. Otherwise, if the integral diverges, the series also diverges.
Therefore, to determine the convergence or divergence of the series ( \sum \frac{1}{n^2} \cos\left(\frac{1}{n}\right) ) from ( n = 1 ) to infinity, we need to evaluate the integral ( \int_{1}^{\infty} \frac{1}{x^2} \cos\left(\frac{1}{x}\right) , dx ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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