Solution Found!
Properties of series Use the properties of
Chapter 10, Problem 9E(choose chapter or problem)
9-14. Properties of series Use the properties of infinite series to evaluate the following series.
\(\sum_{k=0}^{\infty}\left[3\left(\frac{2}{5}\right)^{k}-2\left(\frac{5}{7}\right)^{k}\right]\)
Questions & Answers
QUESTION:
9-14. Properties of series Use the properties of infinite series to evaluate the following series.
\(\sum_{k=0}^{\infty}\left[3\left(\frac{2}{5}\right)^{k}-2\left(\frac{5}{7}\right)^{k}\right]\)
ANSWER:Problem 9E
Properties of series Use the properties of infinite series to evaluate the following series.
Answer;
Step 1 ;
Geometric series ; 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 consisting of n terms is denoted by I , then I = a
Three numbers a, b , c are in G.P . iff =
, i.e . iff
= ac
Sum of first n terms of a G.P = , r
1
Sum of the infinite series of G. P = , r > 1 and
, if -1< r< 1