Kindly sum this series , if its convergent and state why its convergent?

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Answer 1 · 2017-12-30T07:07:52

The given series converges to #5/6#.

We find, #sum_(n=0)^oo(cosnpi)/5^n#,

#=cos0/5^0+cospi/5^1+(cos2pi)/5^2+(cos3pi)/5^3+...,#

#=1-1/5+1/5^2-1/5^3+...,#

So, the series is a geometric series with common ratio

#r=-1/5," and the "1^(st)" term "a=1#.

We know that, # "geometric series converges to "a/(1-r) iff |r| lt 1.#

Consequently, the given series converges to #5/6#.