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Mathematical Expression Editor

Exercises relating to various topics we have studied.

The series is convergentdivergent by limit comparison to the -series with . Likewise, the series is convergentdivergent by limit comparison to the -series with .

Use a Maclaurin series to determine the
dominant behavior of as .

Fill in the blank below with an appropriate constant to make the series absolutely
convergent:

Use the Maclaurin series for and substitute .

Determine whether the series below converges absolutely, conditionally, or diverges.

AbsoluteConditionalDiverge

Show that the terms do not go to zero as .

Determine whether the series below converges absolutely, conditionally, or diverges.

AbsoluteConditionalDiverge

Use the Maclaurin series for and evaluate at . Find the dominant term.

Determine whether the series below converges absolutely, conditionally, or diverges.

AbsoluteConditionalDiverge

Try limit comparison with the harmonic series.

Determine whether the series below converges absolutely, conditionally, or diverges.

AbsoluteConditionalDiverge

Do a Taylor expansion of and evaluate at .

Compute the sum of the series below.

We know the series for .

This means

We conclude

Compute the sum of the series below.

Compute the sum of the series below.

You don’t need Taylor series for this
one.

It’s a telescoping series

Sample Exam Questions

If it converges, find the sum of the series . If the series diverges, explain why.

diverges

We recognize the Taylor series for cosine: The series in question is exactly