MHB Sum of Series $\approx$ Error Estimation

[ineedhelpnow]

use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.

$\sum_{n=1}^{\infty}\frac{1}{3^n+4^n}$

 

[zzephod]

use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.

$\sum_{n=1}^{\infty}\frac{1}{3^n+4^n}$

I will leave you to sum the first ten terms, then we observe:

As $n$ becomes large $3^n \ll 4^n$ so $1/(3^n+4^4)\approx 1/4^n$ and $1/(3^n+4^4) < 1/4^n$ so the remainder after summing the first 10 terms of the series is approximately equal to (and less than) $\int_{10}^{\infty} 1/4^x\; dx$ ...

The above is an upper bound on the truncation error, in this case a lower bound is provided by the first neglected term: $1/(3^{11}+4^{11})$

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