Sucesiones y series

1. Determinate if the succession is converge or diverge.
a) [pic] b) [pic]


2. Find the nth term (general term) an of the succession.
a) [pic] b) [pic]


3. Use theconvergence of a series definition, and determinate if the following series is convergent or divergent, if it converges, calculate its sum.
[pic]


4. Use the integral criteria to determinate ifthe following series is convergent or divergent.
[pic]


5. Determinate if the following series is absolutely convergent.
[pic]


6. Develop the following functions inseries of potency.
a) [pic] b) [pic]


7. a) Find the convergence interval of the following series of potency, and investigate the convergence in the extremes of that interval.
[pic]b) Use the comparison criteria to determinate if the following series converge or diverge.
[pic]


c) Express the following recurring decimal with an infinite series andfind the ration number it represents.
[pic]


DEVELOPMENT


1. a) We need to determinate that[pic]does exist. Then we consider:
[pic]So, [pic], then the succession is convergentand [pic] converges to [pic]


b) We need to determinate that [pic] does exist. Then:
[pic]
So, [pic] is convergent to cero.










2. a) [pic]
Thesuccession is given by the function [pic]. This succession might be written as [pic]




b) [pic]
The succession is given by the function [pic]. This succession might be written as [pic]3. [pic]
The decomposition into partial fractions of the series can be written as [pic]
[pic], so the series is convergent and its sum is [pic]


4. With [pic], where[pic], then
[pic]
Then
[pic]


Next
[pic]
[pic]
Then [pic] converges, therefore it [pic]converges.


5. If we got [pic]

[pic]...