Integral Test for Series Convergence

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Questions and Answers

Under what conditions can the integral test be applied to a series?

The function must be defined for all positive integers.

The function must be continuous, positive, and decreasing. (correct)

The function must be periodic.

The function must have a finite limit as n approaches infinity.

What can be concluded if the integral of a function converges?

The corresponding series converges. (correct)

The function is always increasing.

The function is not continuous.

The corresponding series diverges.

Which of the following is NOT a requirement for the function f(x) used in the integral test?

It must have a finite derivative. (correct)

It must be positive.

It must be continuous.

It must be decreasing.

Which of the following represents the integral test for the convergence of a series?

If the integral converges, then the series converges. Signup and view all the answers

Which of the following series can be tested using the integral test?

$rac{1}{n^2}$ for $n = 1$ to $orever$ Signup and view all the answers

What is the condition for a p-series to be convergent?

p > 1 Signup and view all the answers

When is a geometric series considered divergent?

If the common ratio r is greater than or equal to 1 Signup and view all the answers

Which condition must be met to apply the comparison test for series?

Both series must consist of positive terms Signup and view all the answers

If a_n is less than or equal to b_n for all n, and the sum of b_n is divergent, what can be concluded about a_n?

a_n may be convergent or divergent Signup and view all the answers

In the context of series, what does the term 'convergent' signify?

The sum has a finite limit Signup and view all the answers

What condition must be met for a series to be convergent according to the Alternating Series Test?

The terms must be decreasing and the limit of the terms must approach zero Signup and view all the answers

In the given series ∑(-1)^(n-1) / (n^3 + 1), which test confirms its convergence?

Alternating Series Test Signup and view all the answers

The Ratio Test indicates that a series is absolutely convergent when what condition is satisfied?

The limit ratio is less than 1 Signup and view all the answers

What is the result of the limit lim (n → ∞) n^2 / (n + 1)?

∞ Signup and view all the answers

For the series ∑(e^(-n)/n!), which outcome does the Ratio Test provide?

The series is absolutely convergent Signup and view all the answers

What is a requirement for the terms of a series to be considered decreasing in the context of convergence?

Each term must be less than the next term Signup and view all the answers

What does the Test for Divergence indicate about a series if the limit of its terms is non-zero?

The series diverges. Signup and view all the answers

Which of the following statements about the series ∑(−1)^(n-1) / n is true?

It converges by the Alternating Series Test Signup and view all the answers

For which series is it concluded that it is divergent based on the Test for Divergence?

$ ∑_{n=1}^{ } rac{1 + rac{1}{n}}{n - 1}$ Signup and view all the answers

When applying the Alternating Series Test, which of the following must be shown to confirm convergence?

The absolute value of terms approaches zero and terms are decreasing Signup and view all the answers

What happens to the series $ ∑_{n=1}^{ } rac{2}{n + n}$ as $n$ approaches infinity?

It diverges to infinity. Signup and view all the answers

Which term in the expression $ rac{1}{n + n}$ affects the divergence of the series the most?

The denominator. Signup and view all the answers

What conclusion can be drawn about the limit $ rac{lim a_n}{n - 1}$ if it equals 1?

It indicates divergence. Signup and view all the answers

What does the limit as $t$ approaches infinity of $ an^{-1}(t)$ equal?

$0$ Signup and view all the answers

What does it indicate if $ extstylerac{1}{x^2 + 1}$ is divergent?

The series $ extstyle rac{a_n}{n}$ is divergent. Signup and view all the answers

Which series is shown to be convergent in the content?

$ extstyle rac{(-1)^{n-1}}{n + 1}$ Signup and view all the answers

What is the value of the series $ extstyle rac{1}{2} + rac{1}{4} + rac{1}{4}$?

$rac{3}{4}$ Signup and view all the answers

How can you express the limit of a converging series regarding divergent integrals?

Convergence ensures integral is finite. Signup and view all the answers

Which of the following is true about the series $ extstyle rac{ an^{-1}(x)}{x}$ as $x$ approaches infinity?

It approaches $0$. Signup and view all the answers

If the series $ extstylea_n = (-1)^{n-1} rac{1}{n + 1}$ converges, what can you deduce about the series $ extstylerac{1}{n}$?

It diverges. Signup and view all the answers

Which expression represents the correct evaluation limit for $ extstylerac{ an^{-1}(t)}{t}$ as $t$ approaches infinity?

$0$ Signup and view all the answers

Study Notes

Integral Test

If ( a_n = f(n) ) where ( f(x) ) is continuous, positive, and decreasing on ([1, \infty)), then the series ( \sum a_n ) converges if ( \int f(x) , dx ) converges.
If ( \int f(x) , dx ) diverges, then ( \sum a_n ) is also divergent.
Example shown involves ( \int \frac{1}{x^2 + 1} , dx ) which converges.

Alternating Series Test

For series of the form ( \sum (-1)^{n-1} b_n ):
- Conditions for convergence:
  - ( b_{n+1} \leq b_n ) for all ( n )
  - ( \lim_{n \to \infty} b_n = 0 )
An example series involving ( \frac{1}{n^3 + 1} ) satisfies these conditions and converges.

Ratio Test

If the limit ( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 ), then ( \sum a_n ) is absolutely convergent.
Example involves the series ( \sum \frac{e^{-n}}{n!} ) where the limit condition is met.

p-Series and Geometric Series

A p-series ( \sum \frac{1}{n^p} ) converges if ( p > 1 ) and diverges if ( p \leq 1 ).
A geometric series ( \sum ar^{n-1} ) converges if ( |r| < 1 ) and diverges if ( |r| \geq 1 ).

Comparison Test

To use the comparison test:
- Both series ( a_n ) and ( b_n ) must be positive.
- If ( a_n \leq b_n ) and ( \sum b_n ) converges, then ( \sum a_n ) converges.
- Conversely, if ( a_n \geq b_n ) and ( \sum b_n ) diverges, then ( \sum a_n ) diverges.

Test for Divergence

If ( \lim_{n \to \infty} a_n \neq 0 ), then ( \sum a_n ) is divergent.
Example shown involves ( \sum \frac{2}{n^2} ) leading to divergence based on limit.

General Strategies

Understanding the form of the series can help in choosing appropriate convergence tests.
Recognizing series types can streamline the analysis process for convergence or divergence.

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Description

This quiz tests your understanding of the integral test for determining the convergence or divergence of series. You will explore how the properties of functions and integrals can be applied to series. Prepare to apply these concepts through practical examples.