Does the series converge?If it does,try to find the sum∑n=1→∞ n/2^n 题就是这样~如果有字的话最好是英文

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Does the series converge?If it does,try to find the sum∑n=1→∞ n/2^n 题就是这样~如果有字的话最好是英文

Does the series converge?If it does,try to find the sum∑n=1→∞ n/2^n 题就是这样~如果有字的话最好是英文
Does the series converge?If it does,try to find the sum
∑n=1→∞ n/2^n 题就是这样~如果有字的话最好是英文

Does the series converge?If it does,try to find the sum∑n=1→∞ n/2^n 题就是这样~如果有字的话最好是英文
其实这题可以等化为求和.
Answer:
It does.
∵∑n=1→∞ n/2^n
=1/2^1+2/2^2+3/2^3+...+(n-1)/2^(n-1)+n/2^n,(1)
∴(∑n=1→∞ n/2^n)/2
=1/2^2+2/2^3+3/2^4+...+(n-1)/2^n+n/2^(n+1),(2)
∴(1)-(2),(∑n=1→∞ n/2^n)/2
=1/2+1/2^2+1/2^3+...+1/2^n-n/2^(n+1)
=(1/2)[1-(1/2)^n]/(1-1/2)-n/2^(n+1)
=1-(1/2)^n-n/2^(n+1)
=1-2/2^(n+1)-n/2^(n+1)
=1-(2+n)/2^(n+1),
∴∑n=1→∞ n/2^n =2-(2+n)/2^n
the sum is:
2-(2+n)/2^n.

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