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Chapter 11: Problem 13

Find the interval of convergence of the power series. $$ \sum_{n=0}^{\infty} \frac{n !}{100^{n}} x^{n} $$

Short Answer

Expert verified
The series converges only at \( x = 0 \).

Step by step solution

01

Identify the Power Series

The given power series is \( \sum_{n=0}^{\infty} \frac{n!}{100^n} x^n \). A power series is of the form \( \sum_{n=0}^{\infty} a_n x^n \) where \( a_n = \frac{n!}{100^n} \) in this case.
02

Use the Ratio Test to Find the Radius of Convergence

To find the radius of convergence, apply the ratio test. Calculate \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). Here, \( a_{n+1} = \frac{(n+1)!}{100^{n+1}} \) and \( a_n = \frac{n!}{100^n} \).\[\lim_{n \to \infty} \left| \frac{(n+1)!}{100^{n+1}} \times \frac{100^n}{n!} \right| = \lim_{n \to \infty} \left| \frac{(n+1)}{100} \right| = \lim_{n \to \infty} \frac{n+1}{100} = \infty\]Since the limit is infinity, the radius of convergence is 0.
03

Interpret the Radius of Convergence

The radius of convergence being 0 indicates that the series converges only at \( x = 0 \). This means there is no interval around 0 where the series converges, and it only converges at the center of the series.

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Key Concepts

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

Ratio Test
The Ratio Test is a tool used to determine the convergence of a series. It helps us understand at which points a power series will reach finite sums. For any given series \( \sum a_n \), we look at the limit: \[\lim_{{n \to \infty}} \left|\frac{{a_{n+1}}}{a_n}\right|\] If this limit is:
  • Less than 1, the series converges absolutely.
  • Greater than 1, the series diverges.
  • Equal to 1, the test is inconclusive.
To apply the test to the problem's series, we calculated \( \lim_{n \to \infty} \left| \frac{(n+1)!}{100^{n+1}} \times \frac{100^n}{n!} \right| = \lim_{n \to \infty} \frac{n+1}{100} = \infty \), indicating divergence.
Radius of Convergence
The Radius of Convergence tells us about the interval for which a power series converges. If you have a power series centered at 0, then the series converges if the absolute value of \( x \) is within the radius, \( |x| < R \). In our scenario,
  • The ratio test returned \( \infty \), meaning the radius of convergence, \( R \), is 0.
  • This indicates the series only converges at the center, at \( x = 0 \).
  • If \( R = 0 \), the series doesn't converge anywhere except precisely at the center.
Understanding this concept clarifies where the series behaves nicely and where it doesn't. It is crucial for comprehending how the power series operates across different \( x \) values.
Factorial Series
Factorial Series incorporate factorial terms like \( n! \), making them unique and sometimes complex. A factorial, \( n! \), means \( n \times (n-1) \times (n-2) \ldots \times 1 \), highlighting the rapid growth of terms. When included in a series:
  • The factorial increases much more quickly than exponential terms, \( 100^n \).
  • This rapid growth often results in very small radii of convergence.
In our exercise, \( \frac{n!}{100^n} \) grows too rapidly, leading to the divergence indicated by the ratio test. This growth makes such series tricky, yet fascinating to study.
Power Series Example
Power Series are sums that involve powers of a variable, \( x \). Consider the form \( \sum_{n=0}^{\infty} a_n x^n \). Our specific example is \( \sum_{n=0}^{\infty} \frac{n!}{100^n} x^n \), showcasing a factorial influence. This example is instructive because:
  • It shows how different elements in a series, such as the factorial here, can affect convergence.
  • The factorial caused the radius of convergence to be 0, illustrating how strongly it influences the series behavior.
  • Studying examples like this helps understand the balance between various components within a series.
By looking at more examples, one can better predict how different factors interplay within power series to either allow or prevent convergence.

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Most popular questions from this chapter

If the series is positive-term, determine whether it is convergent or divergent; if the series contains negative terms, determine whether it is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{3^{2 n+1}}{n 5^{n-1}} $$

If a dosage of \(Q\) units of a certain drug is administered to an individual, then the amount remaining in the bloodstream at the end of \(t\) minutes is given by \(Q e^{-c t}\) where \(c>0\). Suppose this same dosage is given at successive \(T\) -minute intervals. (a) Show that the amount \(A(k)\) of the drug in the bloodstream immediately after the \(k\) th dose is given by \(A(k)=\sum_{n=0}^{k-1} Q e^{-n c T}\) (b) Find an upper bound for the amount of the drug in the bloodstream after any number of doses. (c) Find the smallest time between doses that will ensure that \(A(k)\) does not exceed a certain level \(M\) for \(M>Q\)

If the series is positive-term, determine whether it is convergent or divergent; if the series contains negative terms, determine whether it is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{x} \frac{n^{2}-1}{n^{2}+1} $$

Use a power series representation obtained in this section to find a power series representation for \(f(x)\). $$f(x)=x^{3} e^{-x}$$

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{(2 n)^{n}}{\left(5 n+3 n^{-1}\right)^{n}}\)
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