My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum

LinkBack Thread Tools Display Modes
May 7th, 2016, 08:02 PM   #1
Joined: May 2015
From: Earth

Posts: 64
Thanks: 0

Finding an approximation using series

So the problem I am struggling with is attached, and is 8b. I am to use a given power series to find the approximation of a an integral of said power series so that the error between the actual and approximated answer is less than .01 - I got through most of the problem, to the point where I am left with the power series of -1/(1/3)^n, but i'm not sure what to do next.

The answer key (also in the same attached file) tells me that since (1/3)^5 is less than .01 my answer is s sub 4, but I don't understand that.

First, why am I setting (1/3)^n to be less than .01, since my series involves a negative number. And secondly, since it was proven that when n is 5 the error is less than .01, why is my answer s sub 4? Thanks.
eglaud is offline  
May 7th, 2016, 09:14 PM   #2
Math Team
Joined: Dec 2013
From: Colombia

Posts: 7,675
Thanks: 2654

Math Focus: Mainly analysis and algebra
I assume that you mean that the series is $\sum \left(-\frac13\right)^n$ since what you wrote doesn't converge.

Because the series is an alternating series, the series of partial sums $s_n = \sum \limits_{k=1}^n a_k$ has the property $s_1 \gt s_3 \gt \ldots S \ldots \gt s_4 \gt s_2$ (or the reverse - depending on the sign of $s_1$) where $S$ is the limit of the partial sums. The important thing to notice then is that for any $n$ we have
$$s_n \gt S \gt s_{n+1} \quad \text{or} \quad s_{n+1} \gt S \gt s_{n}$$
Thus, you need only find the first $n$ such that $|s_{n}-s_{n+1}| \lt 0.01$.

Now, if $|a_{k+1}| \lt 0.01$ then $|s_{n} - s_{n+1}| = |a_{k+1}| \lt 0.01$, so the job is done.

Last edited by v8archie; May 7th, 2016 at 09:18 PM.
v8archie is offline  

  My Math Forum > College Math Forum > Calculus

approximation, finding, series

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Approximation by geometric series. sepoto Calculus 3 December 5th, 2013 04:26 AM
power series approximation aaron-math Calculus 1 November 28th, 2011 03:04 AM
Quadratic Taylor series approximation Paul4763 Calculus 7 April 29th, 2011 09:54 AM
Maclaurin Series(approximation) naspek Calculus 5 December 10th, 2009 03:27 PM
Approximation of a square root (using a series?) swm Algebra 5 March 30th, 2009 04:26 PM

Copyright © 2019 My Math Forum. All rights reserved.