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Solved problems for the chapter:

Convergent sequence

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Infinite sequence

1. Near point

2. Infinite sequence limit

2.1. Convergence sequence

2.2. Divergent sequence

2.3. Limit of a sequence

3. Evaluation of limits

3.1. Euler's number

4. Forum

View the theory »

Infinite geometric series

1. Infinite sequence

1.1. Sum of an infinite sequence

1.2. Convergence of an infinite geometric sequence

2. Forum

↓ Show theory [2] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:convergent_sequence/5461/

Example 1.

Determine whether the given sequence is convergent:

Example 2.

Determine whether the given sequence is convergent:

Find out and write down the properties (increasing, decreasing, boundedness, convergence) for the given sequence:

Find out which terms of the sequence that have a limit equal to lie in the neighbourhood of .

10.

Determine whether the given sequence is convergent:

11.

Find out and write down the properties (increasing, decreasing, boundedness, divergence) for the given sequence:

12.

Find out and write down the properties (increasing, decreasing, boundedness, convergence) for the given sequence:

13.

Find out which terms of the sequence with limit lie inside .

14.

We have a sequence with limit . Find out which terms of the sequence are located outside the neighbourhood .

15.

We have a sequence with the general term . Calculate the limit of the sequence and determine from which term the values of the terms of the sequence differ from the limit by less than ?

16.

Calculate the limit of the sequence:

17.

Find out and write down the properties (increasing, decreasing, boundedness, divergence) for the given sequence:

18.

We have the sequence . Find the general term of the sequence and determine from which term the values of the terms of the sequence differ from the limit by more than ?

19.

Calculate the limit of the sequence:

20.

We have the sequence . Find the general term of the sequence and determine from which term the values of the terms of the sequence differ from the limit by less than ?

21.

Calculate the sequence limit:

22.

We have a sequence with the general term .

Calculate the limit of the sequence.
Find out from which term the values of the sequence members differ from the limit by less than ?

23.

We have a sequence with the general term .

Prove that the sequence is increasing or decreasing.
Calculate the limit of the sequence.
Determine which sequence terms are located outside the neighbourhood?

24.

For what value of x will the series be convergent. Hence, calculate the sum.

25.

We have the sequence:

Find the general term of the sequence and determine from what term is the values of the terms of the sequence differ from the limit by less than ?

26.

Given:

At what value x will the series converge?
Calculate the sum of the geometric sequence.
For what value of x the sum equal to 45.

27.

We have the sequence:

Find the general term of the sequence and determine from which term the values of the terms of the sequence differ from the limit by less than ?

28.

Given the sequence .

Find the range of values of x for which the given sequence converges?
Calculate the value of x such that the sum of the terms is twice the first term.
Calculate the value of x for which the square of the first term is seven times the fifth term. Round the result to four decimal places.
For x = -1 obtain the sum of the first n terms of the given sequence?
What is the least number of terms that must be added so that the sum will differ from the sum to infinity by less than ?

29.

Given the series:

For what real values of x is the series convergent?
Calculate the value of for which the sum is .
For x = -1, what percentage of the sum of all members of this series is the sum of the first five terms? Round the result to 5 significant figures.
For x = -1, how many members are smaller than ?

30.

Given the real function .

Solve the equation . Give an exact answer.
Prove that .
Evaluate .
Find the range of values of x for which converges, and calculate for which the sum to infinity is 1.