absolute convergence 1

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Question statement

Decide on the convergence of the following series

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			Name	Type	Generated Value

p	integer	7
k	integer	6
t	integer	7
j	integer	4

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$\displaystyle\sum_{n=1}^\infty [\var{p}(-1)^n + \var{k}]\frac{1}{\var{t}n + 1}\left(\frac{1}{\var{j}}\right)^n$

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First of all observe that this is an alternating series.

When $n$ is even we have $\var{p}(-1)^n + \var{k} = \simplify{{p} + {k}}$ , and when $n$ is odd $\var{p}(-1)^n + \var{k} = \simplify{-{p} + {k}}$ .

Therefore, when $n$ is even we have

$[\var{p}(-1)^n + \var{k}] \frac{1}{\var{t}n + 1}\left(\frac{1}{\var{j}}\right)^n = \frac{\simplify{{p} + {k}}}{(\var{t}n+1)\var{j}^n},$

and when $n$ is odd

$[\var{p}(-1)^n + \var{k}] \frac{1}{\var{t}n + 1}\left(\frac{1}{\var{j}}\right)^n = \frac{\simplify{-{p} + {k}}}{(\var{t}n+1)\var{j}^n}.$

Hence

$\left| [\var{p}(-1)^n + \var{k}] \frac{1}{\var{t}n + 1}\left(\frac{1}{\var{j}}\right)^n \right| < \frac{\simplify{{p} + {k}}}{(\var{t}n+1)\var{j}^n}.$

Now, let us look at the series $\displaystyle\sum_{n=1}^\infty \frac{\simplify{{p} + {k}}}{(\var{t}n+1)\var{j}^n}$ . Observe that

$\frac{\simplify{{p} + {k}}}{(\var{t}n+1)\var{j}^n} < \frac{\simplify{{p} + {k}}}{\var{j}^n}.$

But we know that $\displaystyle\sum_{n=1}^\infty \frac{\simplify{{p} + {k}}}{\var{j}^n}$ is convergent (In fact one needs to show this. But it can be done easily).

Then by Comparison Test, $\displaystyle\sum_{n=1}^\infty \frac{\simplify{{p} + {k}}}{(\var{t}n+1)\var{j}^n}$ is convergent.

That in turn implies, again by comparison test, that $\displaystyle\sum_{n=1}^\infty \left| [\var{p}(-1)^n + \var{k}] \frac{1}{\var{t}n + 1}\left(\frac{1}{\var{j}}\right)^n \right|$ is convergent.

Hence, $\displaystyle\sum_{n=1}^\infty [\var{p}(-1)^n + \var{k}] \frac{1}{\var{t}n + 1}\left(\frac{1}{\var{j}}\right)^n$ is absolutely convergent. So, it is convergent.

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This question is used in the following exam:

Series Homework 1 by Ugur Efem in Ugur's workspace.

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Ugur Efem 1 year, 6 months ago

Ugur Efem 1 year, 6 months ago

Ugur Efem 1 year, 6 months ago

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absolute convergence 1

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Feedback

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Ugur Efem 1 year, 6 months ago

Ugur Efem 1 year, 6 months ago

Ugur Efem 1 year, 6 months ago

Error in variable testing condition

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Generated value: integer

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Daniella's copy of True or False

Generated value: `integer`