Chapter 1: Problem 22
Use the ratio test to find whether the following series converge or diverge: $$\sum_{n=1}^{\infty} \frac{10^{n}}{(n !)^{2}}$$
Short Answer
Expert verified
The series converges because the limit \( L = 0 < 1 \).
Step by step solution
01
Identify the general term
The general term of the series is given by \(a_n = \frac{10^n}{(n!)^2}\).
02
Apply the ratio test formula
Calculate the ratio \(L = \frac{a_{n+1}}{a_n}\). Compute \(\frac{a_{n+1}}{a_n} = \frac{\frac{10^{n+1}}{[(n+1)!]^2}}{\frac{10^n}{(n!)^2}}\).
03
Simplify the ratio
Simplify the expression: \[ \frac{a_{n+1}}{a_n} = \frac{10^{n+1} \times (n!)^2}{10^n \times [(n+1)!]^2} = \frac{10 \times 10^{n} \times (n!)^2}{10^n \times (n+1)^2 \times (n!)^2} = \frac{10}{(n+1)^2} \].
04
Compute the limit
Find the limit of the ratio as \(n \to \infty\): \[ L = \frac{10}{(n+1)^2} \to 0 \].
05
Determine convergence or divergence
Since \(L < 1\) (it approaches 0), the series converges by the ratio test.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
infinite series
An infinite series represents the sum of an infinite sequence of terms. Understanding infinite series is crucial in many areas of mathematics and science.
It can be written in the summation notation \(\[\begin{equation} \sum_{n=1}^{\infty} a_n \end{equation}\]\), where \(a_n\) are the terms of the series.
In these series, the number of terms is unbounded, potentially leading to meaningful insights into the behavior of functions and systems.
For the series we are examining, \(\[\begin{equation} \sum_{n=1}^{\text{infty}} \frac{10^n}{(n!)^2} \end{equation}\]\), using the ratio test helps us determine if the sum converges or diverges.
It can be written in the summation notation \(\[\begin{equation} \sum_{n=1}^{\infty} a_n \end{equation}\]\), where \(a_n\) are the terms of the series.
In these series, the number of terms is unbounded, potentially leading to meaningful insights into the behavior of functions and systems.
For the series we are examining, \(\[\begin{equation} \sum_{n=1}^{\text{infty}} \frac{10^n}{(n!)^2} \end{equation}\]\), using the ratio test helps us determine if the sum converges or diverges.
ratio test
The ratio test is a popular method for determining the convergence of an infinite series.
This test analyzes the limit of the ratio of consecutive terms in a series.
For a series \( \sum_{n=1}^{\infty} a_n \), we calculate:
\[\[\begin{equation} L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n}\right| \end{equation}\]\]. If \( L < 1\), the series converges.
If \( L > 1\) or if \( L \to \infty\), the series diverges.
When \(L = 1\), the test is inconclusive.
In our exercise, the ratio test helps us find the convergence of \(\[\begin{equation} \sum_{n=1}^{\text{infty}} \frac{10^n}{(n!)^2} \end{equation}\]\). By applying the ratio test formula, simplification leads us to the limit as \(L \to 0\), which indicates convergence.
This test analyzes the limit of the ratio of consecutive terms in a series.
For a series \( \sum_{n=1}^{\infty} a_n \), we calculate:
\[\[\begin{equation} L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n}\right| \end{equation}\]\]. If \( L < 1\), the series converges.
If \( L > 1\) or if \( L \to \infty\), the series diverges.
When \(L = 1\), the test is inconclusive.
In our exercise, the ratio test helps us find the convergence of \(\[\begin{equation} \sum_{n=1}^{\text{infty}} \frac{10^n}{(n!)^2} \end{equation}\]\). By applying the ratio test formula, simplification leads us to the limit as \(L \to 0\), which indicates convergence.
convergence and divergence
Convergence and divergence are essential concepts in analyzing infinite series.
When a series converges, the sum of its infinite terms approaches a finite value.
When a series diverges, the sum of its terms grows without bounds or does not settle to a single value.
For example, the harmonic series \(\[\begin{equation} \sum_{n=1}^{\infty} \frac{1}{n} \end{equation}\]\) diverges, whereas the geometric series \( \sum_{n=1}^{\infty} (1/2)^n \) converges to 1.
In our problem, using the ratio test on \(\[\begin{equation} \sum_{n=1}^{\text{infty}} \frac{10^n}{(n!)^2} \end{equation}\]\), leads us to the conclusion that the series converges because the limit calculated is less than 1.
When a series converges, the sum of its infinite terms approaches a finite value.
When a series diverges, the sum of its terms grows without bounds or does not settle to a single value.
For example, the harmonic series \(\[\begin{equation} \sum_{n=1}^{\infty} \frac{1}{n} \end{equation}\]\) diverges, whereas the geometric series \( \sum_{n=1}^{\infty} (1/2)^n \) converges to 1.
In our problem, using the ratio test on \(\[\begin{equation} \sum_{n=1}^{\text{infty}} \frac{10^n}{(n!)^2} \end{equation}\]\), leads us to the conclusion that the series converges because the limit calculated is less than 1.
factorials
Factorials are fundamental in combinatorics and analysis. Denoted by \( n! \), the factorial of a number \( n \) is the product of all positive integers up to \( n \).
So, \( n! = n \times (n-1) \times ... \times 1 \).
To see how factorials work, let's consider \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
In our exercise, factorials are used in the series term \( \[\begin{equation} a_n = \frac{10^n}{(n!)^2} \end{equation}\] \).
While applying the ratio test, understanding factorials is critical as it simplifies the ratio expression. In our solution, noticing that \( (n+1)! = (n+1) \times n! \) allows us to cancel terms efficiently, leading to a more straightforward calculation.
So, \( n! = n \times (n-1) \times ... \times 1 \).
To see how factorials work, let's consider \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
In our exercise, factorials are used in the series term \( \[\begin{equation} a_n = \frac{10^n}{(n!)^2} \end{equation}\] \).
While applying the ratio test, understanding factorials is critical as it simplifies the ratio expression. In our solution, noticing that \( (n+1)! = (n+1) \times n! \) allows us to cancel terms efficiently, leading to a more straightforward calculation.
limit calculation
Limit calculations are integral in calculus, especially when determining series convergence using tests like the ratio test.
Limits help us understand the behavior of functions or sequences as the input approaches a given value, often infinity.
For a function \(f(x)\), the limit as \(x \to c\) is the value that \(f(x)\) approaches as \(x\) gets closer to \(c\).
In our problem, during the ratio test, we calculate:
\( L = \lim_{n \to \infty} \frac{10}{(n+1)^2} = 0 \). Because the limit \(L \) is less than 1, it confirms the series \( \sum_{n=1}^{\text{infty}} \frac{10^n}{(n!)^2}\} \) converges. The calculation of this limit involves understanding how factorials influence the terms as \( n \to \infty \).
Limits help us understand the behavior of functions or sequences as the input approaches a given value, often infinity.
For a function \(f(x)\), the limit as \(x \to c\) is the value that \(f(x)\) approaches as \(x\) gets closer to \(c\).
In our problem, during the ratio test, we calculate:
\( L = \lim_{n \to \infty} \frac{10}{(n+1)^2} = 0 \). Because the limit \(L \) is less than 1, it confirms the series \( \sum_{n=1}^{\text{infty}} \frac{10^n}{(n!)^2}\} \) converges. The calculation of this limit involves understanding how factorials influence the terms as \( n \to \infty \).
