The discussion focuses on estimating the sum of the series $\sum_{n=1}^{\infty}\frac{1}{3^n+4^n}$ by calculating the first 10 terms and evaluating the error. It is established that for large values of $n$, the term $1/(3^n+4^n)$ approximates to $1/4^n$. The upper bound for the truncation error is determined using the integral $\int_{10}^{\infty} 1/4^x\; dx$, while the lower bound is given by the first neglected term, $1/(3^{11}+4^{11})$. This method provides a clear framework for error estimation in infinite series.
PREREQUISITESMathematicians, students studying calculus, and anyone interested in series convergence and error estimation techniques.
ineedhelpnow said:use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.
$\sum_{n=1}^{\infty}\frac{1}{3^n+4^n}$