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Chapter 1: Q25P (page 1)

Question: Find the first few terms of the Maclaurin series for each of the following functions and check your results by computer. 2Xe2x-1

Short Answer

Expert verified

Hence, the required series is: .

Step by step solution

01

Maclaurin Series

Maclaurin series is basically a type of power series expansion of the function f(x) about the origin, with all the terms having positive values expanded as:

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Most popular questions from this chapter

∑n=1∞ (sin⁡x)n(−1)n2n

Use power series to evaluate the function at the given point. Compare with computer results, using the computer to find the series, and also to do the problem without series. Resolve any disagreement in results (see Example 1).earcsinx+In(1-xe)atx=0.0003.

By computer or tables, find the exact sum of each of the following series.

(a)∑n=1∞n(4n2-1)2(b)∑n=1∞n3n!(c)∑n=1∞n(n+1)3n

(a) Using computer or tables (or see Chapter 7,Section 11),verify that∑n=1∞(1/n2)=π26=1.6449=,and also verify that the error in approximating the sum of the series by the first five terms is approximately 0.1813.

(b) By computer or tables verify that ∑n=1∞(1/n2)(1/2)n=π212-(1/2)(ln2)2=0.5815+

the sum of the first five terms is0.5815+

(c) Prove theorem (14.4). Hint: The error is |∑N+1∞anxn|.

Use the fact that the absolute value of a sum is less than or equal to the sum of the absolute values. Then use the fact that |an+1|≤|an|to replace all anby aN+1 , and write the appropriate inequality. Sum the geometric series to get the result.

Show that24-x=1+18xwith an error less than132for 0<x<1. Hint: Let x=4y , and use the theorem (14.4) .
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