SOLUTION: Find the sum of the series 1+(1+a)/2!+(1+a+a^2)/3!+....infinity

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Question 1089120: Find the sum of the series
1+(1+a)/2!+(1+a+a^2)/3!+....infinity
Answer by ikleyn(52890) About Me  (Show Source):
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Let S = 1+(1+a)/2!+(1+a+a^2)/3!+....infinity


Multiply all and every terms/term by (1-a). You will get

(1-a)*S = %281-a%29 + %281-a%5E2%29%2F2%21 + %281-a%5E3%29%2F3%21 + . . . = 

        = 1+%2B+1%2F2%21+%2B+1%2F3%21+%2B+.+.+.+%29 - (a + a%5E2%2F2%21 + a%5E3%2F3%21 + . . .) = 

        = e - %28e%5Ea-1%29 = 1+%2B+e+-+e%5Ea.


Hence,  S = %281+%2B+e+-+e%5Ea%29%2F%281-a%29, under the condition a =/= 1.


        At a = 1, then the sum S = 1 + e.

Solved.