How do I find the sum of the infinite series 1 + #1/5# + #1/25# +... ?

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↳Redirected from "How do I change #int_0^1int_0^sqrt(1-x^2)int_sqrt(x^2+y^2)^sqrt(2-x^2-y^2)xydzdydx# to cylindrical or spherical coordinates?"

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Answer 1 · 2018-05-27T13:10:59

#5/4#

You are looking for

#S = \sum_{i=0}^\infty \frac{1}{5^n}= \sum_{i=0}^\infty (\frac{1}{5})^n#

In general,

#S = \sum_{i=0}^\infty a^n#

converges if and only if #|a|<1#, which is obviously our case. If #|a|<1#, the sum evaluates to

#S = \sum_{i=0}^\infty a^n = \frac{1}{1-a}#

So, in your case,

#S = \sum_{i=0}^\infty (\frac{1}{5})^n = \frac{1}{1-1/5} = \frac{1}{4/5} = 5/4#