Question

Determine whether the series is convergent or divergent.

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Question

Determine whether the series is convergent or divergent.

First of all, we try to find out the summation series of the series.

Then, we use the Integral Test here which is:

• If \(\int_1^\infty f(x) \, dx\) convergent, then \(\sum_{n=1}^{\infty} a_n\) is convergent.

• If \(\int_1^\infty f(x) \, dx\) convergent, then \(\sum_{n=1}^{\infty} a_n\) is divergent.

Then you basically take the \(\lim_{m \to \infty} \int_1^m f(x) \, dx\) which in this case is \(\frac{1}{n^2 + n^3}\).

You proceed to take the integral with Power Rule and then finally taking the limit.

Since the limit exists, it converges which is the final answer.

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Question

Determine whether the series is convergent or divergent.

We use the Test for Divergence where we take the summation as \(\lim_{k \to \infty} ke^{-k}\) which is \(0\).

It converges since the limit is 0.

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Question

Determine whether the series is convergent or divergent.

We use the Integral Test here which is:

• If \(\int_2^\infty f(x) \, dx\) convergent, then \(\sum_{n=2}^{\infty} a_n\) is convergent.

• If \(\int_2^\infty f(x) \, dx\) convergent, then \(\sum_{n=2}^{\infty} a_n\) is divergent.

Then you basically take the \(\lim_{m \to \infty} \int_2^m f(x) \, dx\) which in this case is \(\frac{1}{n \ln n}\).

You proceed to take the integral with U-Substitution and then finally taking the limit.

Since the limit does not equal to 0, it diverges which is the final answer.

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Question

Determine whether the series is convergent or divergent.

Solutions

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Question

Determine whether the series is convergent or divergent.

The power-P rule of the Integral Test is applied!

FIX: The Integral Test will actually work in this case!

Solutions

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Question

Determine whether the series is convergent or divergent.

First of all, we try to find out the summation series of the series.

Then, we use the Integral Test here which is:

• If \(\int_1^\infty f(x) \, dx\) convergent, then \(\sum_{n=1}^{\infty} a_n\) is convergent.

• If \(\int_1^\infty f(x) \, dx\) convergent, then \(\sum_{n=1}^{\infty} a_n\) is divergent.

Then you basically take the \(\lim_{m \to \infty} \int_1^m f(x) \, dx\) which in this case is \(\frac{\sqrt{n} + 4}{n^2}\).

You proceed to take the integral with Power Rule and then finally taking the limit.

Since the limit exists, it converges which is the final answer.

Solutions

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Question

Determine whether the series is convergent or divergent.

First of all, we try to find out the summation series of the series.

Then, we use the Integral Test here which is:

• If \(\int_0^\infty f(x) \, dx\) convergent, then \(\sum_{n=0}^{\infty} a_n\) is convergent.

• If \(\int_0^\infty f(x) \, dx\) convergent, then \(\sum_{n=0}^{\infty} a_n\) is divergent.

Then you basically take the \(\lim_{m \to \infty} \int_0^m f(x) \, dx\) which in this case is \(\frac{1}{3+4x}\).

You proceed to take the integral with U-Substitution and then finally taking the limit.

Since the limit does not equal to 0, it diverges which is the final answer.

Solutions

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Question

Determine whether the series is convergent or divergent.

The power-P rule of the Integral Test is applied!

Solutions

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Question

Determine whether the series is convergent or divergent.

We use the Integral Test here which is:

• If \(\int_1^\infty f(x) \, dx\) convergent, then \(\sum_{n=1}^{\infty} a_n\) is convergent.

• If \(\int_1^\infty f(x) \, dx\) convergent, then \(\sum_{n=1}^{\infty} a_n\) is divergent.

Then you basically take the \(\lim_{m \to \infty} \int_1^m f(x) \, dx\) which in this case is \(\frac{1}{n^{\sqrt{2}}}\).

You proceed to take the integral with Power Rule and then finally taking the limit.

Since the limit exists, it converges which is the final answer.

The power-P rule of the Integral Test can also be applied here!

Solutions

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