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Chapter 8: Problem 43

Find all values of \(p\) such that \(\sum_{k=1}^{\infty} \frac{p^{k}}{k}\) converges.

Short Answer

Expert verified
The series converges for all values of \(p\) in the open interval (-1, 1).

Step by step solution

01

Define the Series

The first step is preparing the series given in the problem: \(\sum_{k=1}^{\infty} \frac{{p^{k}}}{{k}}\)
02

Apply the Ratio Test

Next, apply the ratio test. If the limit as \(k\) approaches infinity of the absolute ratio of the \(k+1\) term to the \(k\) term of the series is less than 1, the series converges. This gives \(\lim_{k\to\infty} \left\lvert \frac{{p^{k+1}/(k+1)}}{{p^k/k}} \right\rvert = \lim_{k\to\infty} \left\lvert \frac{p}{k+1}\cdot k \right\rvert\). Simplifying, we get \(|p|<1\).
03

Identify the Range of Convergence

The last step is to identify the range of values of p that satisfy the condition found in Step 2, which is \(-1<p<1\). Hence, the series is convergent for all values of \(p\) in the open interval \((-1, 1)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Ratio Test
The ratio test is a handy tool in mathematics to determine whether an infinite series converges or diverges. Imagine you have a series; the ratio test involves looking at the ratio of successive terms. If this ratio is consistently less than 1, the series converges.

Here鈥檚 how it works:
  • Take any term \( a_k \) in the series, and then find the next term \( a_{k+1} \).
  • Calculate the ratio \( R = \left| \frac{a_{k+1}}{a_k} \right| \). It鈥檚 essential to consider the absolute value.
  • Finally, determine the limit of this ratio as \( k \) approaches infinity: \( \lim_{k \to \infty} R \).
If this limit is less than 1, the series converges. If the limit is more than 1, or if it does not exist, the test is inconclusive. This test is especially good for series involving exponential terms, such as \( p^k \), because ratios of exponential terms are easy to compute.
Exploring Infinite Series
An infinite series is just a sum of an infinite sequence of numbers. Think of it as adding up numbers forever. In mathematical terms, an infinite series looks like this: \( \sum_{k=1}^{\infty} a_k \), where \( a_k \) represents each term in the series.

But why do we care if an infinite series converges? Convergence tells us if this seemingly endless addition results in a finite number. Here are some points about convergence:
  • Convergence means that as you keep adding more and more numbers from the series, the total sum approaches a particular limit.
  • Divergence, on the other hand, means that the sum keeps getting larger and never settles on a number.
Understanding whether an infinite series converges is crucial in many mathematics and physics applications, as it determines whether we have a best answer or need a better approach.
Introduction to Geometric Series
A geometric series is a special type of series where each term is a fixed multiple of the previous term. This repeated multiplication factor is called the common ratio \( r \). A geometric series looks something like: \( a + ar + ar^2 + ar^3 + \ldots \).

Here's what you need to know about geometric series:
  • If the absolute value of the common ratio \( |r| < 1 \), the geometric series converges.
  • Conversely, if \( |r| \ge 1 \), the series diverges. This is crucial because a convergent series allows us to calculate the sum.
  • The sum of an infinite geometric series when \( |r| < 1 \) can be calculated using the formula \( S = \frac{a}{1-r} \), where \( a \) represents the first term.
Geometric series are prevalent in various fields, from calculating interest compounding to solving certain physics problems, making them essential for students to grasp fully.

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Most popular questions from this chapter

Define $$a_{n}=\sum_{k=1}^{n} \frac{1}{n+k}$$. By thinking of \(a_{n}\) as a Riemann sum, identify the definite integral to which the sequence converges.

Define the sequence \(a_{n}\) with \(a_{1}=\sqrt{3}\) and \(a_{n}=\sqrt{3+2 a_{n-1}}\) for \(n \geq 2 .\) Show that \(\left\\{a_{n}\right\\}\) converges and estimate the limit of the sequence.

Use a Taylor polynomial to estimate \(\int_{0}^{\pi} \frac{\sin x}{x} d x\) accurate to within 0.00001 . (This value will be used in the next section.)

Determine the radius and interval of convergence. $$\sum_{k=0}^{\infty} \frac{k}{4^{k}} x^{k}$$

(a) use a Taylor polynomial of degree 4 to approximate the given number, (b) estimate the error in the approximation and (c) estimate the number of terms needed in a Taylor polynomial to guarantee an accuracy of \(10^{-10}\) $$\sqrt{1.1}$$
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