What is the interval of convergence of #sum (2+2/n)^n((x-1)/2)^n#?

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Answer 1 · 2016-08-01T10:39:17

#sum_{n=1}^oo (2+2/n)^n((x-1)/2)^n le e/(2-x)# for #0 < x < 2#

#sum_{n=1}^oo (2+2/n)^n((x-1)/2)^n = sum_{n=1}^oo (1+1/n)^n(x-1)^n #

but

#e = lim_{n->oo} (1+1/n)^n ge (1+1/n)^n #

so

#sum_{n=1}^oo (2+2/n)^n((x-1)/2)^n le e sum_{n=1}^oo(x-1)^n#

supposing now #abs(x-1) < 1# we have

# sum_{n=1}^oo(x-1)^n equiv lim_{n->oo}((x-1)^n-1)/(x-1-1) = 1/(2-x)#

Finally

#sum_{n=1}^oo (2+2/n)^n((x-1)/2)^n le e/(2-x)# for #abs(x-1)<1#