# Algebra II : Infinite Series

## Example Questions

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### Example Question #1 : Infinite Series

Which of the following infinite series has a finite sum?

Explanation:

For an infinite series to have a finite sum, the exponential term (the term being raised to the power of  in each term of the series) must be between  and .  Otherwise, each term is larger than the previous term, causing the overall sum to grow without bounds towards infinity.

### Example Question #2 : Infinite Series

Evaluate:

The series diverges.

Explanation:

The sum of an infinite series , where , can be calculated as follows:

Setting :

### Example Question #3 : Infinite Series

Evaluate:

The series diverges

The series diverges

Explanation:

An infinite series  converges to a sum if and only if . However, in the series , this is not the case, as . This series diverges.

### Example Question #4 : Infinite Series

Evaluate:

The series diverges

Explanation:

The sum of an infinite series , where , can be calculated as follows:

Setting :

### Example Question #5 : Infinite Series

Evaluate:

Explanation:

Write the formula for infinite geometric series.

The value of  is the first term of the series, which is .

The value of the common ratio, , is also .

The ratio is  because if we were to write out the first few terms in the series we would see,

each term is three fourths more than the previous term therefore, giving us the ratio.

Substitute the values into the equation and evaluate.

### Example Question #6 : Infinite Series

If  and , what will be the sum of the infinite series?

Explanation:

Write the infinite series formula.

Substitute the values of  and .

### Example Question #7 : Infinite Series

Evaluate the infinite series for

Explanation:

The first term of this sequence is 10. To find the common ratio r, we can just divide the second term by the first: . So "r" is -0.9. We can find the infinite sum using the formula where a is the first term and r is the common ratio:

### Example Question #8 : Infinite Series

Find the sum of the infinite series

Cannot be determined - the sum is infinite

Cannot be determined - the sum is infinite

Explanation:

An infinite sum is only calculable if where r is the common ratio. We can find the common ratio easily by dividing the second term by the first: . This is greater than 1, so we can't find the infinite sum - it is infinite.

### Example Question #9 : Infinite Series

What is the sum?

Explanation:

Write the formula to find the sum of an infinite geometric series.

The first term is: .

The common ratio is:

Substitute the values into the formula.

Rewrite the complex fraction.

The sum will converge to .

### Example Question #10 : Infinite Series

What is the sum?

Explanation:

Write the formula for the sum of an infinite series.

The value  is the first term, and  is the common ratio.

Divide the second term with the first term, third term and the second, and so forth, and we will get a common ratio of:

Substitute the values into the formula.

Rewrite the complex fraction using a division sign.

Change the division sign to a multiplication and take the reciprocal of the second term.

The series will converge to .

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