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Chapter 11: Problem 35

Determine the values of \(p\) for which the given series converges. \(\sum_{n=1}^{\infty} \frac{1}{p^{n}}\)

Short Answer

Expert verified
The series converges for \( p > 1 \).

Step by step solution

01

Identify the Type of Series

The given series is \( \sum_{n=1}^{\infty} \frac{1}{p^n} \). This is a geometric series where each term is in the form \( ar^n \) with \( a = \frac{1}{p} \) and \( r = \frac{1}{p} \).
02

Determine the Convergence Condition for Geometric Series

A geometric series \( \sum_{n=0}^{\infty} ar^n \) converges if the absolute value of the common ratio \( |r| < 1 \). For our series, the common ratio is \( r = \frac{1}{p} \).
03

Apply the Convergence Condition

Apply the convergence condition \( \left| \frac{1}{p} \right| < 1 \). This results in the inequality \( \frac{1}{p} < 1 \), which simplifies to \( p > 1 \).
04

Conclusion

The given series converges if \( p > 1 \). This is because, for \( p > 1 \), the terms of the series diminish to zero and the sum of the infinite geometric series exists.

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Key Concepts

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

Convergence Condition
When dealing with an infinite series, particularly a geometric series, it is important to understand under what conditions the series converges. The convergence condition is a criterion that indicates when the sum of an infinite series approaches a finite value.
  • For a geometric series, the series converges if the absolute value of the common ratio is strictly less than 1.
  • This means that each term becomes smaller and smaller, eventually approaching zero, which allows the series to sum up to a finite number.
Understanding this condition is crucial since not all series converge. If a series does not converge, it means that its sum becomes infinite as more terms are added.
Geometric Series
A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant, called the common ratio. It has the general form: \[ S = a + ar + ar^2 + ar^3 + \dots\]
  • In our exercise, the first term \(a\) and the common ratio \(r\) are both equal to \( \frac{1}{p} \).
  • The series looks like \( \sum_{n=1}^{\infty} \frac{1}{p^n} \), indicating it's a geometric series.
Geometric series are particularly important because under certain conditions, they provide convenient formulas for calculating the sum of infinite terms. This characteristic makes them a popular choice for solving problems involving repeated multiplication.
Infinite Series
An infinite series is a sum of terms that continues indefinitely. Mathematically, it is expressed as \( \sum_{n=1}^{\infty} a_n \). In simple terms, you keep adding numbers forever.
  • Some infinite series, like a geometric series, can sum up to a finite number, which we call convergence.
  • Others grow without bounds and thus diverge, meaning they do not sum to a finite limit.
Whenever you encounter an infinite series, the first task is to determine whether it converges. The study of infinite series involves understanding these properties to know when they will add up neatly and when they will not.
Common Ratio
The common ratio is a key factor in determining the behavior of a geometric series. It is the fixed number you multiply by to get from one term to the next in a sequence.
  • In our problem, the common ratio is \( \frac{1}{p} \).
  • If this ratio satisfies the condition \(|r| < 1\), the series will converge.
The value of the common ratio helps decide if the series will have a finite sum or not. It is crucial to examine this ratio to determine the series' convergence. For any geometric series, carefully checking the ratio helps tutor you in understanding whether or not you can assign a finite total to the ongoing sum of its terms.

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Most popular questions from this chapter

Deduce from the arctangent series (Example 11 ) that $$\pi=\frac{6}{\sqrt{3}} \sum_{n=0}^{\infty} \frac{(-1)^{n}}{2 n+1}\left(\frac{1}{3}\right)^{n}$$ Then use this alternating series to show that \(\pi=3.14\) accurate to two decimal places.

Use power series rather than I'H么pital's rule to evaluate the given limit. \(\lim _{x \rightarrow 0} \frac{1-\cos x}{x\left(e^{x}-1\right)}\)

Find the interval of convergence of each power series. \(\sum_{n=0}^{\infty}(5 x-3)^{n}\)

Compare your estimates with the exact values given by a computer algebra system. \(\int_{0}^{1 / 2} \frac{1}{1+x^{2}+x^{4}} d x\)

Find the interval of convergence of each power series. \(\sum_{n=1}^{\infty} \frac{(-4)^{n} x^{n}}{\sqrt{2 n+1}}\)
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