If the constant p in the general p-series is replaced by avariable x for x > 1, then the

Chapter 9, Problem 63

(choose chapter or problem)

If the constant \(p\) in the general \(p\)-series is replaced by a variable \(x\) for \(x > 1\), then the resulting function is called the Riemann zeta function and is denoted by

                 \(\zeta(x)=\sum_{k=1}^{\infty} \frac{1}{k^{x}}\)

(a) Let \(S_{n}\) be the nth partial sum of the series for \(\zeta\) (3.7). Find \(n\) such that \(S_{n}\) approximates \(\zeta\) (3.7) to two decimal-place accuracy, and calculate sn using this value of \(n\). [Hint: Use the right inequality in Exercise 36(b) of Section 9.4 with \(f(x)=1 / x^{3.7}\).]

(b) Determine whether your CAS can evaluate the Riemann zeta function directly. If so, compare the value produced by the CAS to the value of \(S_{n}\) obtained in part (a).

Equation Transcription:

Text Transcription:

p

x

x > 1,

zeta (x) sum_k=1 ^infinity 1/k^x

s sub n

zeta

n

f(x)=1/x^3.7