Determining Basic Convergence

This last section will be focusing on some basic tips that can help you determine the convergence of an infinite series.

The first tip is called the test for divergence.

The second tip is called the integral test. Remember how we proved the convergence of an infinite series in the last section by comparing it with the improper integral of the related function? Well, this leads us to the next convergence test.

Finally, the last tip for this section is to go through the sequence of partial sums. Keep adding the terms of the sequence together to see what the value of the infinite series approaches.

Example 2.4.1

Determine if the infinite series

converges or diverges.

Example 2.4.2

Use the integral test to determine the convergence of the infinite series

Example 2.4.3

A frog leaps forward one foot and then leaps forward half the distance it did in the previous jump. If this frog were to continue leaping forever, how far would it end up? The data for how far the frog leaps, in feet, for each trial is given in the sequence

an = , with starting index n=1

We want to see if the infinite series converges or diverges.

Going through the test for divergence,

= = 0.

Since the sequence converges to 0, this means that the infinite series can still converge.

Next, we want to take a look at {Sn}, or the sequence of partial sums. Finding the partial sums...

S1 = a1 = 1

Example 2.4.4

Determine the convergence of the infinite series

Chapter Problems

1. If the next term of a sequence is either always greater than or always less than the previous term, then it is known as what type of sequence?
2. Find the common difference for the sequence

{1, -5, -11, -17, ...}

3. Find the common ratio for the sequence

4. For questions 4-9, do the following sequences converge or diverge? If it converges, find its limit.

{3n2-n}n=1

5. n=1
6. {2-ne-n}n=1
7. n=1
8. n=2
9. n=1
10. Find the 4th partial sum of the sequence

n=1

11. Find the 5th partial sum of the sequence

n=1

12. For questions 12-19, determine if the series converges or diverges.

13.

14.

1. ${\sum }_{n=3}^{\infty }\frac{1}{n\ln n\ln \left(\ln n\right)}$

   Clarification: this web problem begins at n=3. The source begins at n=1, where its logarithmic term is undefined; n=3 also gives a positive continuous function for the integral-test proof.

2. 
3. 
4. 
5. 

ANSWERS

1. Monotonic sequence
2. common difference = -6
3. common ratio = 3
4. To determine the convergence, the limit must be evaluated.

= lim (3n2-n)

=

=

= ∞

Since the limit goes to infinity, the sequence diverges.

5. To determine the convergence, the limit must be evaluated.

=

=

=

= *

= 1 * 1

= 1

Since the limit is 1, the sequence converges to 1.

6. To determine the convergence, the limit must be evaluated.

= (2-ne-n)

=

= 0

Since the limit is 0, the sequence converges to 0.

7. To determine the convergence, the limit must be evaluated.

=

= 0

Since the limit is 0, the sequence converges to 0.

8. To determine the convergence, the limit must be evaluated.

=

= ln(1)

= 0

Since the limit is 0, the sequence converges to 0.

9. To determine the convergence, the limit must be evaluated.

=

=

As the limit is , you can apply L'Hôpital’s rule to take the new limit

=

= ∞

Since the limit goes to infinity, the sequence diverges.

10. To find the 4th partial sum, the first four terms need to be evaluated.

a1 = 1

a2 =

a3 =

a4 = ,

S4 = a1 + a2 + a3 + a4

= 1 + + +

=

1. To find the 5th partial sum, the first five terms need to be evaluated.

a1 = = 2

a2 = =

a3 = = 1

` a4 = =

a5 = = ,

S5 = a1 + a2 + a3 + a4 + a5

= 2 + + 1 + +

=

1. First put the series through the test for divergence.

Since it is a geometric sequence where r = , it converges to

0 and passes the test for divergence.

Going through the sequence of partial sums, the series

converges to 3.

1. First put the series through the test for divergence.

=

=

= 1

Since the limit of the series terms is 1 and not 0, the series diverges.

1. The series converges through the integral test.

Letting u = ln n...

=

=

=

=

=

=

Because there is a value for the improper integral, the series converges.

1. 1. For $x\ge 3$, the function $f\left(x\right)=\frac{1}{x\ln x\ln \left(\ln x\right)}$ is positive and continuous. Its denominator $g\left(x\right)=x\ln x\ln \left(\ln x\right)$ has derivative $g\prime \left(x\right)=\ln x\ln \left(\ln x\right)+\ln \left(\ln x\right)+1>0$, so $f$ is decreasing.
   2. Let $u=\ln \left(\ln x\right)$, so $du=\frac{dx}{x\ln x}$. Then ${\int }_3^b\frac{dx}{x\ln x\ln \left(\ln x\right)}={\int }_{\ln \left(\ln 3\right)}^{\ln \left(\ln b\right)}\frac{du}{u}=\ln \left(\ln \right(\ln b\left)\right)-\ln \left(\ln \right(\ln 3\left)\right)$. Both logarithmic arguments are positive for $b\ge 3$.
   3. As $b\to \infty$, $\ln \left(\ln \right(\ln b\left)\right)\to \infty$, so the improper integral diverges. The integral test therefore proves that the displayed tail starting at $n=3$ diverges.

16. The series diverges through the integral test.

=

=

= (ln b - ln 1)

= ln b

= ∞

Since the value of the improper integral is infinity, the series diverges.

1. The series diverges.

3n = ∞

The series terms do not converge to 0 and therefore diverges.

1. The series converges through the integral test.

Letting u = 3x+2...

=

=

=

=

=

=

=

Because there is a value for the improper integral, the series converges.

1. The series diverges.

3n+2 = ∞

The series terms do not converge to 0 and therefore diverges.