MAT 21C Lecture 3: MAT 21C – Lecture 3 – Convergence of Geometric and Telescoping Series

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges 2.3 3.4 41)n+2) What is the formula for the nth partial sum of the series? What is the sum of the series? Select the correct choice below and, if necessary,il in the answer box to complete your answer 0 A. The sum of the series is O B. The series diverges (Type an integer or a simplified fraction )

-เก้า) Findthesumofthese ries n=1 The sum of the series is Type an exact answer, using radicals as needed.)

Σ ln( n +1 Determine whether the series n'42 | converges or diverges. If it converges, find its sum. Select the correct choice below and, if necessary, fill in the answer box within your choice ○ A. O B. The series diverges because it is a geometric series with Irl 21 (n-1)-0 The sumoftheseriesis The series converges because lim Itn (Type an exact answer.) The series converges because it is a geometric series with Irl - 1. The sum of the series is on 0. The sum of the series is (Type an exact answer.) (n+1)-0 or fails to exist ( ) doesnot 0 D. The series diverges because lim In 0 E. The series diverges because lim In s because lim ndoes not exist n + 2

4. (8 points) Use the Direct Comparison Test to determine if the series converges or diverges. n3 1 n=1 5. (12 points) An infinite series is given by 13n +1 3n4 Find an expression for the nth partial sum S of the series. Then find lim S and determine whether the series converges or diverges. If the series converges, find its sum. n=1

(7) In this problem we will use Integral Test to estimate the sum S of the series by the n-th partial sum Sn-> ak. We define the remainder, Rn to be Rn-S-Sn- Theorem (Error Estimate for Integral Test) Suppose that f(k) = ak for k = 1, 2, f is continuous positive decreasing function and f (x) dx converges. Then, ak k=n+1 , where where ecreasingfunction and「(r)dr coivngsrkril f(x) dx