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Example Questions
Example Question #2771 : Algebra Ii
Determine the sum of:
Notice that this an infinite series.
Find the common ratio by dividing the second term with the first term, third term with the second term, and so forth. The common ratio should be same for each term divided.
The common ratio is:
Write the formula for the sum of an infinite series.
The answer is:
Example Question #14 : Infinite Series
Determine the sum of the infinite series:
To find the sum, write the formula for infinite series.
Determine the common ratio by dividing the second term with the first term, or third term with the second term. They should have similar common ratios.
Simplify the complex fractions to verify that both have similar common ratios.
Substitute the known terms into the formula.
Simplify the complex fraction.
The sum will converge to .
Example Question #111 : Mathematical Relationships And Basic Graphs
Calculate the sum of the following series, if it exists:
Each successive term in this series is determined by multiplying the previous term by ; hence, the series is geometric and its common ratio is .
Since , the sum of this series exists and we can calculate it via the following formula:
,
where is the desired sum of the geometric series, is the common ratio of the geometric series, and is the first term appearing in the series.
Here, and . Hence, the sum of this geometric series is
.
Example Question #11 : Infinite Series
Determine the sum:
Write the formula for the sum of an infinite series.
The is the common ratio between the terms in the series. This can be found by dividing the second term with the first term, the third term with the second, and so forth.
Substitute the known terms into the equation.
Simplify the complex fraction.
The answer will converge to:
Example Question #111 : Summations And Sequences
Determine the sum, if possible:
Write the formula for the sum of an infinite series.
The first term is two. To determine the common ratio, we will need to divide the second term by the first, third term with the second, and so forth. The common ratio should be same for each term.
The common ratios are verified to be the same. Substitute the into the formula. This value must be between negative one and one or the series will diverge!
Simplify this complex fraction.
The series will converge to .
Example Question #12 : Infinite Series
Determine the sum, if possible:
Write the formula for the sum of an infinite series.
Determine the common ratio, . Divide the second term by the first. This ratio should be the same if the third term was divided by the second term, and so forth.
The common ratio is .
Substitute the known terms into the equation.
Solve the complex fraction.
The answer is:
Example Question #111 : Mathematical Relationships And Basic Graphs
Find the sum, if possible:
Write the formula for infinite series.
Determine the common ratio by dividing the second term with the first term.
The ratio should also be similar if the third term is divided by the second term. Verify that the common ratios are similar.
Substitute the known terms into the equation.
Simplify the complex fraction.
The sum will converge to:
Example Question #20 : Infinite Series
Determine the sum, if possible:
Find the common ratio of the infinite series. Divide the second term with the first, or the third with the second term. The common ratios should be similar to each other.
Write the formula for infinite series and substitute the numbers.
Simplify the denominator.
The sum converges to .
Example Question #21 : Infinite Series
Determine the sum, if possible:
Determine the common ratio of the infinite series by dividing the second term with the first term and the third term with the second term. The common ratios should be similar.
Write the formula for infinite series, and substitute the terms.
Simplify the complex fraction.
The answer is:
Example Question #22 : Infinite Series
Determine the sum of:
Notice that there is a pattern only after the second term. Rewrite the infinite series.
We can find the common difference among the second and third term, and with the third term and fourth term, but there is no relation with the first and second term.
To determine the common difference, we will need to divide the third term with the second term or the fourth term with the third.
Write the formula for the infinite series.
The first term is NOT 1! The first term is the starting value of the series, which is . Substitute the terms.
This means that:
The answer is:
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