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Remainder term Consider the geometric series , which has
Chapter 11, Problem 73AE(choose chapter or problem)
Remainder term Consider the geometric series \(S=\sum_{k=0}^{\infty} r^{k}\), which has the value 1/(1 - r) provided |r| < 1, and let \(S_{n}=\sum_{k=0}^{n-1} r^{k}=\frac{1-r^{n}}{1-r}\) be the sum of the first n terms. The remainder \(R_{n}\) is the error in approximating S by \(S_{n}\). Show that
\(R_{n}=\left|S-S_{n}\right|=\left|\frac{r^{n}}{1-r}\right|\)
Questions & Answers
QUESTION:
Remainder term Consider the geometric series \(S=\sum_{k=0}^{\infty} r^{k}\), which has the value 1/(1 - r) provided |r| < 1, and let \(S_{n}=\sum_{k=0}^{n-1} r^{k}=\frac{1-r^{n}}{1-r}\) be the sum of the first n terms. The remainder \(R_{n}\) is the error in approximating S by \(S_{n}\). Show that
\(R_{n}=\left|S-S_{n}\right|=\left|\frac{r^{n}}{1-r}\right|\)
ANSWER:Problem 73 AERemainder term Consider the geometric series , which has the value 1/(1 r) provided |r|<1, and let be the sum of the first n terms. The remainder Rn Is the error in approximating S by Sn. Show that .Answer; Step-1; A sequence ( finite or infinite ) of non zero numbers is called a geometric progression ( abbreviated G.P) iff the ratio of any terms to its preceding term is constant . This non zero constant is usually denoted by ‘r’ and is called common ratio. General term of G.P is = a Thus , if ‘a’ is the first term and ‘r’ is the common ratio , then the G.P is a , ar , a,a………….according as it is finite or infinite.Remarks ; If the last term of a G.P