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Question

express the sum using summation notation

-1+(2/2)+(-3/4)+(4/8)+...

Answer 1

express the sum using summation notation

-1+(2/2)+(-3/4)+(4/8)+...

The numerator series is –1, 2, 3, 4, …
So it can be written as: N*(-1)^N
N = 1, 2, 3, ...

The denominator series is: 1, 2, 4, 8,…
So it can be written as: 2^J
J = 0, 1, 2, 3, ...

We want them to start with the same N, so the second series can be written as: 1/2*(2, 4, 8, 16,...)
So it can be written as: 1/2*(2^N)
N = 1, 2, 3, ...

Combining the above (that is divide the 1sr by the 2nd):

Series = 2*Sum[ N*(-1/2)^N ] ,where N goes from 1 to infinity.

Answer 2

summation from i=1 to n of (-1)^n *(n/2^(n-1))

Answer 3

[k=0, ∞]∑(k+1)/(-2)^k

Answer 4

for numerator: -1, -3, -5, -7 ..... = summation from k = 1 to infinity {-2k-1}
2,4,6, summation k= 1 to inifnity {2k}
denominatior:, 2,4,8.... = sumamtion of k = 1 to infity {2^k * k!}

[-2k-1][2k]
---------------> simplified form!!!!!! (trust me this is the right answer!)
[2^k * k!]

Answer 5

numerator series is -1,2,-3,4.. = (-1^n)*n
denominator series is 1,2,3,8 which is a geometric progression and is equal to 2^(n-1)

thus the series will be ((-1^n)*n)/(2^(n-1))