Question

Euler found the sum of the p-series with p = 4: (4) = sum infinity n=1 1/n4 = pi4/90 Use Euler's result to find the sum of the series. a) sum infinity n=1 (3/n)4 b) sum ininity k=6 1/(k-3)4

Answer 1

The complete answer to this question is considerably more complex than it may first appear. I will outline the very basics, and further in-depth analysis is certainly at MSc/PhD level mathematics.

The very basic and quick answer is that the infinite sum of a p-series is given by the Riemann zeta function .

Where:

ζ(s)=∞∑n=11ns

Some specific solutions are:

ζ(2)=∞∑n=11n2=π26 (the Bessel Problem)

ζ(3)=∞∑n=11n3=1.20205... (Apéry's constant )

ζ(4)=∞∑n=11n4=π490

Answer 2

Part a

∞∑n=1(3n)4=∞∑n=134n4=81∞∑n=11n4=81π490∑n=1∞(3n)4=∑n=1∞34n4=81∑n=1∞1n4=81π490

Part b

∞∑k=61(k−3)4=∞∑k=31((k+3)−3)4[Drop the index from k=6 to k=3]=∞∑k=31k4=∞∑k=11k4−114−124=π490−1716