The CSAT and Practice[+] are designed by the Climb Foundation to help candidates. We are advocates for more opportunity to shine and less opportunity to fail, and we strive to level the playing field.

# Sample Test 4 Question 6

Let $$F_k=F_{k-1}+F_{k-2}$$ where $$F_0=0, F_1=1$$. Define the series $$s(x)= \sum_{k=1}^\infty F_k / x^k$$ for $$x>2$$.

Show that $$s(x)=\frac{x}{x^2-x-1}$$.

The above links are provided as is. They are not affiliated with the Climb Foundation unless otherwise specified.

## Hints

• Hint 1
Use the recursive definition of $$F_k$$ to split $$s(x)$$ into two sums. Carefully treat the base cases.
• Hint 2
Try rewriting the summation in terms of $$s(x).$$
• Hint 3
You may relabel the index of each sum in order to get it to match the definition of $$s(x).$$

## Solution

The key is to breakdown $$F_k$$ using its recursive definition, then relabelling the indices to get $$s(x)$$ again. The first term of the sum $$\frac{F_1}{x}$$ cannot be broken down as it is a base case, and must be taken out of the sum first. $$$\begin{split} s(x)&=\sum_{k=1}^\infty \frac{F_k}{x^k}\\ &=\frac{F_1}{x} + \sum_{k=2}^\infty \frac{F_k}{x^k} \\ &=\frac{F_1}{x} + \sum_{k=2}^\infty \frac{F_{k-1}}{x^k} + \sum_{k=2}^\infty \frac{F_{k-2}}{x^k} \\ &=\frac{F_1}{x} + \sum_{i=1}^\infty \frac{F_{i}}{x^{i+1}} + \bigg(\frac{F_0}{x^2} + \sum_{j=1}^\infty \frac{F_{j}}{x^{j+2}}\bigg) \\ &=\frac{1}{x} + \sum_{i=1}^\infty \frac{F_{i}}{x^{i+1}} + \bigg(\frac{0}{x^2} + \sum_{j=1}^\infty \frac{F_{j}}{x^{j+2}}\bigg) \\ &=\frac{1}{x} + \frac{1}{x}\sum_{i=1}^\infty \frac{F_{i}}{x^i} + \frac{1}{x^2} \sum_{j=1}^\infty \frac{F_{j}}{x^j} \\ &=\frac{1}{x} + \frac{1}{x}s(x) + \frac{1}{x^2}s(x) \\ \end{split}$$$

Finally, we rearrange to get $$s(x)=\frac{x}{x^2-x-1}$$.

If you have queries or suggestions about the content on this page or the CSAT Practice Platform then you can write to us at . Please do not write to this address regarding general admissions or course queries.