Find the smallest number n of terms needed to obtain an approximation of the series

Question

Find the smallest number n of terms needed to obtain an approximation of the series

∞

∑10/k⁴ accurate to 10^-4

k=1

Answer 1

∑_k=1^∞ (10/k⁴) = 10 ∑_k=1^∞ (1/k⁴)

This sum notation ∑_k=1^∞ (1/k⁴) is what we called the Riemann Zeta Function ζ(4) = π⁴/90.

Riemann Zeta Function has several proofs in proofwiki.org and it shows 5 proofs of ζ(4) = π⁴/90. If you need one single proof, analysis and computation, just search for it there and get the one appropriate in your study.

ζ(4) = π⁴/90 ≈ 1.08232323371

For ζ(4), the least number of terms accurate to the ten thousandth digit (10^-4) is 17 (n=17)

1/1^4 + 1/2^4 +1/3^4 +...+1/16^4 ≈1.0822

1/1^4 + 1/2^4 +1/3^4 +...+1/16^4 + 1/17^4 ≈1.0823

But for ∑_k=1^∞ (10/k⁴) the least number of terms accurate to the ten thousandth digit (10^-4) is 34 (n=34)

10•1/1^4 + 10•1/2^4 +10•1/3^4 +...+10•1/33^4 ≈10.8231

10•1/1^4 + 10•1/2^4 +10•1/3^4 +...+10•1/33^4 + 10•1/34^4 ≈10.8232

*Make sure you use a calculator or computing app with the summation function to make it a lot easier to solve.