THEOREM 5.10 Remainder Estimate from the Integral Test 2 n=1 Suppose an is a convergent series with positive terms. Suppose there exists a function f satisfying the following three conditions: i. f is continuous, ii. f is decreasing, and iii. f(n) = an for all integers n > 1. 00 Let Sy be the Nth partial sum of an. For all positive integers N, n=1 1 SN + + N N+1 n=1 +8Î 69 f(a) de < an <$x+[ºf(x) dx. Σ 0 Voo In other words, the remainder RN = an Sn = Σ an satisfies the following (n=N+1 n=1 estimate: Sa f()dx < Rx < Sºf()dx . (5.10) N+1 This is known as the remainder estimate. 6. Use Theorem 5.10 < (Section 5.3 in Vol. 2 of OpenStax Calculus) for this problem. 1 How many terms of the series would you need to add to n=2 n=2 n(In n)3 find the value of the series with an error less than 0.01?