I've got in my assignment to show if the following series converges or diverges.
∞∑n=1(e1n−1−1n)
Attempt:
∞∑n=1(e1n−1−1n)=∞∑n=1(1+1n+12!n2+13!n3+...−1−1n)=∞∑n=1(12!n2+13!n3+...)=∞∑n=1(1n+12!n2+13!n3+...)
At this point I'm lost. I tried using D'Alambert as follows:
liman+1an=limn+1√e−1−1n+1n√e−1−1n
which I tried to simplify with basic limit laws (hopefully correctly):
limn+1√e−1n√e−1=1
I don't know where to go from here. Thank you for all your help in advance.
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