To determine The first four terms and four partial sums of the series ∑ n = 1 ∞ ( 1 n − 1 n + 1 ) and check whether the series appear to be convergent or not. Explanation Consider the infinite series ∑ n = 1 ∞ ( 1 n − 1 n + 1 ) . Obtain the first four terms of the sequence by substituting n = 1 , 2 , 3 and 4 in the expression 1 n − 1 n + 1 . a 1 = 1 1 − 1 1 + 1 = 1 − 1 2 = 1 2 Similarly, when n = 2 the sequence becomes, a 2 = 1 2 − 1 2 + 1 = 1 2 − 1 3 = 3 − 2 6 = 1 6 When n = 3 the sequence becomes, a 3 = 1 3 − 1 3 + 1 = 1 3 − 1 4 = 4 − 3 12 = 1 12 Further, when n = 4 the sequence becomes, a 4 = 1 4 − 1 4 + 1 = 1 4 − 1 5 = 5 − 4 20 = 1 20 Therefore, the first four terms a 1 , a 2 , a 3 and a 4 of the given sequence are 1 2 , 1 6 , 1 12 and 1 20 _ , respectively. Let the partial sum S n be the sum of first n terms of the series, that is, S n = ∑ k = 1 n ( 1 k − 1 k + 1 ) . Simply the above expression as follows. S n = ∑ k = 1 n ( 1 k − 1 k + 1 ) = ∑ k = 1 n ( k + 1 − k ( k ) ( k + 1 ) ) = ∑ k = 1 n ( 1 ( k ) ( k + 1 ) ) Obtain the first four partial sums of the above series as follows

11th Edition

ISBN: 9780134437705

Author: Washington

Publisher: PEARSON

See similar textbooks

Concept explainers

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.

Videos

Question

Chapter 30.1, Problem 42E

To determine

The first four terms and four partial sums of the series ∑n=1∞(1n−1n+1) and check whether the series appear to be convergent or not.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 30.1, Problem 41E

Chapter 30.1, Problem 43E

Students have asked these similar questions

(a) Suppose a function f: C→C has an isolated singularity at wЄ C. State what it means for this singularity to be a pole of order k. (2 marks) (b) Let f have a pole of order k at wЄ C. Prove that the residue of f at w is given by 1 res (f, w): = Z dk (k-1)! >wdzk−1 lim - [(z — w)* f(z)] . (5 marks) (c) Using the previous part, find the singularity of the function 9(z) = COS(πZ) e² (z - 1)²' classify it and calculate its residue. (5 marks) (d) Let g(x)=sin(211). Find the residue of g at z = 1. (3 marks) (e) Classify the singularity of cot(z) h(z) = Z at the origin. (5 marks)

1. Let z = x+iy with x, y Є R. Let f(z) = u(x, y) + iv(x, y) where u(x, y), v(x, y): R² → R. (a) Suppose that f is complex differentiable. State the Cauchy-Riemann equations satisfied by the functions u(x, y) and v(x,y). (b) State what it means for the function (2 mark) u(x, y): R² → R to be a harmonic function. (3 marks) (c) Show that the function u(x, y) = 3x²y - y³ +2 is harmonic. (d) Find a harmonic conjugate of u(x, y). (6 marks) (9 marks)

Please could you provide a step by step solutions to this question and explain every step.

Chapter 30 Solutions

Basic Technical Mathematics

Ch. 30.1 - Prob. 1PE Ch. 30.1 - Prob. 2PE Ch. 30.1 - Prob. 1E Ch. 30.1 - Prob. 2E Ch. 30.1 - Prob. 3E Ch. 30.1 - Prob. 4E Ch. 30.1 - Prob. 5E Ch. 30.1 - Prob. 6E Ch. 30.1 - Prob. 7E Ch. 30.1 - Prob. 8E

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.