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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 $n = 0$.
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 $3$ and which sums to $6$. If no such series exists, enter N/A in both spaces below:
• Give an example of a geometric series whose first term is $3$ and which sums to $2$. If no such series exists, enter N/A in both spaces below:
• Give an example of a geometric series whose first term is $3$ and which sums to $1$. 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 $n=3$ 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

$\ln 2$ $\ln 3$ $\ln 4$ $\ln 5$ $\ln 6$ $\ln 7$