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Mathematical Expression Editor

Exercises relating to fundamental properties of series.

Compute the sum of the series:

Split the series into two separate geometric series
and evaluate each separately.

Compute the sum of the series:

One way to proceed is to reindex the series so that
it starts at .

Compute the sum of the series:

Simplify as much as possible and write in standard
form for a geometric series.

Give an example of a geometric series whose first term is and which sums
to . If no such series exists, enter N/A in both spaces below:

Give an example of a geometric series whose first term is and which sums
to . If no such series exists, enter N/A in both spaces below:

Give an example of a geometric series whose first term is and which sums
to . If no such series exists, enter N/A in both spaces below:

Find the correct value of the constant in the blank below which makes the sum of
the series equal to zero:

One reasonable strategy is to separate the series into two
geometric series and reindex to apply the standard formula.

Compute the sum of the infinite series below:

There is a closely-related telescoping
series:

Compute the sum of the infinite series below.

Expand the terms using partial
fractions and compute several partial sums by hand. What you get is something like
a telescoping series, but cancellations occur in a slightly different way than usual.

The formula for a general partial sum is

Use your answer to the previous exercise to compute the sum of the series:

Use the answer you just found to compute the sum of the series:

To drop the
term, you can just calculate it and then subtract it from the previous
series.

Sample Quiz Questions

Compute the exact value of the infinite series

The series is not a geometric series or Taylor series, we compute the first few partial
sums: In particular, writing the sum of logarithms as a logarithm of a product leads
to substantial cancellation. By letting , we get .