Mathematics

Problem

Suppose

p>1

and let

R_{N}(p)=\sum_{n=N+1}^{\infty}\frac1{n^{p}}

be the remainder (error) incurred when the first

N

terms of a convergent

p

-series are used to approximate its infinite sum. (a) Prove that

R_{N}(p)\leq\displaystyle\int_{N}^{\infty}\frac{dx}{x^{p}}

for every integer

N\ge1

. (b) Evaluate the improper integral in part (a) and hence obtain a formula that bounds

R_{N}(p)

above in terms of

N

and

p

. State your result clearly. (c) Specialize your bound to the case

p=2

. Determine the smallest integer

N

for which the partial sum

\displaystyle\sum_{n=1}^{N}\frac1{n^{2}}

approximates the infinite series to within

0.01