Explanation Given: The given series are: 1. ∑ s = 2 ∞ s ( s + 1 ) s 2 + 1 x s − 1 2. ∑ p = 1 ∞ p 2 ( p + 1 ) ! x p + 4 Definition used: Power series: “The general power series is an infinite series of the form, ( 1 ) ∑ m = 0 ∞ a m ( x − x 0 ) m = a 0 + a 1 ( x − x 0 ) + a 2 ( x − x 0 ) 2 + ⋯ . Here, x is a variable. a 0 , a 1 , a 2 , ⋯ are constants, called the coefficients of the series. x 0 is a constant, called the center of the series. In particular, if x 0 = 0 , a power series in powers of x ( 2 ) ∑ m = 0 ∞ a m x m = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ⋯ .” Calculation: Consider the first series ∑ s = 2 ∞ s ( s + 1 ) s 2 + 1 x s − 1 . Here, the power under the summation sign is x s − 1 . To convert the power under the summation sign be x m , shift m = s − 1 so that s = m + 1 . Shift s = m + 1 in the series. ∑ s = 2 ∞ s ( s + 1 ) s 2 + 1 x s − 1 = ∑ m + 1 = 2 ∞ ( m + 1 ) ( m + 1 + 1 ) ( m + 1 ) 2 + 1 x m + 1 − 1 = ∑ m = 1 ∞ ( m + 1 ) ( m + 2 ) ( m + 1 ) 2 + 1 x m Check the result by writing first few terms explicitly. ∑ s = 2 ∞ s ( s + 1 ) s 2 + 1 x s − 1 = ∑ m = 1 ∞ ( m + 1 ) ( m + 2 ) ( m + 1 ) 2 + 1 x m 2 ( 2 + 1 ) ( 2 ) 2 + 1 x 2 − 1 + 3 ( 3 + 1 ) ( 3 ) 2 + 1 x 3 − 1 + ⋯ = ( 1 + 1 ) ( 1 + 2 ) ( 1 + 1 ) 2 + 1 x 1 + ( 2 + 1 ) ( 2 + 2 ) ( 2 + 1 ) 2 + 1 x 2 + ⋯ 2 ( 3 ) 5 x 1 + 3 ( 4 ) 10 x 2 + ⋯ = ( 2 ) ( 3 ) 5 x 1 + ( 3 ) ( 4 ) 10 x 2 + ⋯ 6 5 x + 12 10 x 2 + ⋯ = 6 5 x + 12 10 x 2 + ⋯ Hence, the result is true

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 5.1, Problem 15P

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To shift: The index of the given series so that the power under the summation sign is xm and check the result by writing first few terms explicitly.

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Chapter 5.1, Problem 14P

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Chapter 5 Solutions

Advanced Engineering Mathematics

Ch. 5.1 - WRITING AND LITERATURE PROJECT. Power Series in...Ch. 5.1 - Determine the radius of convergence. Show the...Ch. 5.1 - Determine the radius of convergence. Show the...Ch. 5.1 - Determine the radius of convergence. Show the...Ch. 5.1 - Determine the radius of convergence. Show the...Ch. 5.1 - Apply the power series method. Do this by hand,...Ch. 5.1 - Apply the power series method. Do this by hand,...Ch. 5.1 - Apply the power series method. Do this by hand,...Ch. 5.1 - Apply the power series method. Do this by hand,...Ch. 5.1 - Find a power series solution in powers of x. Show...

Ch. 5.1 - Find a power series solution in powers of x. Show...Ch. 5.1 - Find a power series solution in powers of x. Show...Ch. 5.1 - Find a power series solution in powers of x. Show...Ch. 5.1 - Find a power series solution in powers of x. Show...Ch. 5.1 - Prob. 15P Ch. 5.1 - Prob. 16P Ch. 5.1 - CAS PROBLEMS. IVPs Solve the initial value problem...Ch. 5.1 - Prob. 18P Ch. 5.1 - Prob. 19P Ch. 5.2 - Legendre functions for n = 0. Show that (6) with n...Ch. 5.2 - Legendre functions for n = 1. Show that (7) with n...Ch. 5.2 - Special n. Derive (11′) from (11). Ch. 5.2 - Prob. 4P Ch. 5.2 - Obtain P6 and P7. Ch. 5.2 - Prob. 11P Ch. 5.2 - Prob. 12P Ch. 5.2 - Rodrigues’s formula. Obtain (11′) from (13). Ch. 5.2 - Prob. 14P Ch. 5.2 - Prob. 15P Ch. 5.3 - Prob. 1P Ch. 5.3 - Prob. 2P Ch. 5.3 - Prob. 3P Ch. 5.3 - Prob. 4P Ch. 5.3 - Prob. 5P Ch. 5.3 - Prob. 6P Ch. 5.3 - Prob. 7P Ch. 5.3 - Prob. 8P Ch. 5.3 - Prob. 9P Ch. 5.3 - Prob. 10P Ch. 5.3 - Find a basis of solutions by the Frobenius method....Ch. 5.3 - Find a basis of solutions by the Frobenius method....Ch. 5.3 - Find a general solution in terms of hypergeometric...Ch. 5.3 - Find a general solution in terms of hypergeometric...Ch. 5.3 - Find a general solution in terms of hypergeometric...Ch. 5.3 - Find a general solution in terms of hypergeometric...Ch. 5.3 - Find a general solution in terms of hypergeometric...Ch. 5.3 - Find a general solution in terms of hypergeometric...Ch. 5.4 - Prob. 1P Ch. 5.4 - Prob. 2P Ch. 5.4 - Prob. 3P Ch. 5.4 - Prob. 4P Ch. 5.4 - Prob. 5P Ch. 5.4 - Prob. 6P Ch. 5.4 - Prob. 7P Ch. 5.4 - Prob. 8P Ch. 5.4 - Prob. 9P Ch. 5.4 - Prob. 10P Ch. 5.4 - Prob. 11P Ch. 5.4 - Prob. 12P Ch. 5.4 - Prob. 13P Ch. 5.4 - Prob. 14P Ch. 5.4 - Interlacing of zeros. Using (21) and Rolle’s...Ch. 5.4 - Prob. 16P Ch. 5.4 - Bessel’s equation. Show that for (1) the...Ch. 5.4 - Elementary Bessel functions. Derive (22) in...Ch. 5.4 - Prob. 19P Ch. 5.4 - Prob. 20P Ch. 5.4 - Use the powerful formulas (21) to do Probs. 19–25....Ch. 5.4 - Prob. 22P Ch. 5.4 - Use the powerful formulas (21) to do Probs. 19–25....Ch. 5.4 - Use the powerful formulas (21) to do Probs. 19–25....Ch. 5.4 - Use the powerful formulas (21) to do Probs. 19–25....Ch. 5.5 - Prob. 1P Ch. 5.5 - Prob. 2P Ch. 5.5 - Prob. 3P Ch. 5.5 - Prob. 4P Ch. 5.5 - Prob. 5P Ch. 5.5 - Prob. 6P Ch. 5.5 - Prob. 7P Ch. 5.5 - Prob. 8P Ch. 5.5 - Prob. 9P Ch. 5.5 - Hankel functions. Show that the Hankel functions...Ch. 5.5 - Modified Bessel functions of the first kind of...Ch. 5.5 - Prob. 13P Ch. 5.5 - Reality of Iv. Show that Iv(x) is real for all...Ch. 5.5 - Modified Bessel functions of the third kind...Ch. 5 - Prob. 1RQ Ch. 5 - What is the difference between the two methods in...Ch. 5 - Prob. 3RQ Ch. 5 - Prob. 4RQ Ch. 5 - Write down the most important ODEs in this chapter...Ch. 5 - Can a power series solution reduce to a...Ch. 5 - What is the hypergeometric equation? Where does...Ch. 5 - List some properties of the Legendre polynomials. Ch. 5 - Prob. 9RQ Ch. 5 - Can a Bessel function reduce to an elementary...Ch. 5 - POWER SERIES METHOD OR FROBENIUS METHOD Find a...Ch. 5 - POWER SERIES METHOD OR FROBENIUS METHOD Find a...Ch. 5 - POWER SERIES METHOD OR FROBENIUS METHOD Find a...Ch. 5 - Prob. 14RQ Ch. 5 - Prob. 15RQ Ch. 5 - Prob. 16RQ Ch. 5 - Prob. 17RQ Ch. 5 - Prob. 18RQ Ch. 5 - Prob. 19RQ Ch. 5 - Prob. 20RQ

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Chapter 5 Solutions