How do you use the integral test to determine if #1/3+1/5+1/7+1/9+1/11+...# is convergent or divergent?
There are other summations that would work here as well, but this will suffice.
So, we will evaluate the following integral:
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To use the integral test to determine if the series ( \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} + \ldots ) is convergent or divergent, follow these steps:

Consider the function ( f(x) = \frac{1}{x} ). This function is positive, continuous, and decreasing for ( x \geq 3 ), which includes all terms in the series.

Integrate ( f(x) ) from 3 to infinity: [ \int_{3}^{\infty} \frac{1}{x} , dx = \ln(x) \bigg{3}^{\infty} = \lim{b \to \infty} (\ln(b)  \ln(3)) ]

Evaluate the limit: [ \lim_{b \to \infty} (\ln(b)  \ln(3)) = \infty ]

Since the integral diverges, by the integral test, the series ( \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} + \ldots ) also diverges.
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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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