Question

1^-4 + 2^-4 + 3^-4 + 4^-4 + ...
------------------------------ =
1^-4 + 3^-4 + 5^-4 + 7^-4 + ...

How to solve this?

Answer 1

let the Top part is S, and the bottom part is T, so the question is similar to simplifying of S/T

Answer 2

S/T = (1^-4 + 2^-4 + 3^-4 + 4^-4 + ...) / (1^-4 + 3^-4 + 5^-4 + 7^-4 + ... )

Answer 3

we can grouping into two part of the value of S,
S = (1^-4 + 3^-4 + 5^-4 + 7^-4 + ...)+(2^-4 + 4^-4 + 6^-4 + 8^-4 + ...)

Answer 4

simplify that fraction and then it can be
S/T = 1 + (2^-4 + 4^-4 + 6^-4 + 8^-4 + ...)/(1^-4 + 3^-4 + 5^-4 + 7^-4 + ...)

Answer 5

factorize that 2^-4 + 4^-4 + 6^-4 + 8^-4 + ... = 2^-4(1^-4 + 2^-4 + 3^-4 + 4^-4 + ...) = 1/16 (1^-4 + 2^-4 + 3^-4 + 4^-4 + ...)

Answer 6

now we have the equation :
S/T = 1 + 1/16 (S + T)

Answer 7

opssss i mean, it will be
S/T = 1 + 1/16 (S+T)/T

Answer 8

multiple by 16 to both side give us :
16S/T = 16 + (S + T)/T
16S/T = 16 + S/T + 1
16 S/T - S/T = 17

Answer 9

so the answer is 17?

Answer 10

16 S/T - S/T = 17
(16 - 1) S/T = 17
15 S/T = 17
S/T = 17/15

Answer 11

Tq very much