Chapter 2: Q13P (page 57)

Test each of the following series for convergence.
$\sum_{} \frac{{(1 + i)}^{n}}{{(2 = i)}^{n}}$

Short Answer

Expert verified

The series converges, i.e., $ρ < 1$ .

Step by step solution

Given Information.

The given series, i.e., $\sum_{} \frac{{(1 + i)}^{n}}{{(2 = i)}^{n}}$

Definition of Convergent and Divergent series.

A convergent series is one in which the partial sums all gravitate to the same finite number, also known as a limit. Divergent refers to any series that is not converging.

Calculate the value of ρn.

Find the value of $a_{n}$ and $a_{n + 1}$

$a_{n} = {(\frac{1 + i}{2 - i})}^{n} . . . (1) a_{n + 1} = {(\frac{1 + i}{2 - i})}^{n + 1} . . . (2)$

If $ρ < 1$ series converges, if $ρ > 1$ then diverges.

ρ = \lim_{n \to \infty} ρ_{n}

Test the series for convergence.

Use the ratio test.

$ρ_{n} = \frac{a_{n + 1}}{a_{n}} = {(\frac{1 + i}{2 - i})}^{n + 1 - n} = (\frac{1 + i}{2 - i})$

Calculate the value of $ρ$ , i.e.,

$ρ = \lim_{n \to \infty} |ρ_{n}| = \lim_{n \to \infty} |\frac{1 + i}{2 - i}| = |0.2 + 0.6 i| = 0.632$

Hence, the series converges, i.e., $ρ < 1$ .

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Test each of the following series for convergence.
$\sum_{} \frac{{(1 + i)}^{n}}{{(2 = i)}^{n}}$

Short Answer

Step by step solution

Given Information.

Definition of Convergent and Divergent series.

Calculate the value of ρn.

Test the series for convergence.

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Test each of the following series for convergence.∑(1+i)n(2=i)n

Short Answer

Step by step solution

Given Information.

Definition of Convergent and Divergent series.

Calculate the value of ρn.

Test the series for convergence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

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Study anywhere. Anytime. Across all devices.

Test each of the following series for convergence.
$\sum_{} \frac{{(1 + i)}^{n}}{{(2 = i)}^{n}}$