2006 iTest Problems/Problem 31
Problem
The value of the infinite series can be expressed as where and are relatively prime positive numbers. Compute .
Solution
Notice that , and notice that the numerator contains . Also, notice that . We can rewrite the expression by factoring and splitting the fractions. In the fraction , we can split the fraction as , making a telescoping series. Plugging in values results in .
In the fraction , plugging in the first few values results in . The structure suggests that we could try to model the same fraction with a telescoping series.
Substituting for in results in . Also, because , we find that the expression is equal to , confirming that is a telescoping series. The expression equals .
Therefore, , so .
See Also
2006 iTest (Problems, Answer Key) | ||
Preceded by: Problem 30 |
Followed by: Problem 32 | |
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