Sum the given series to infinity.

Question

Sum to infinity

$$1 + \frac34 + \frac{3 \cdot 5}{4 \cdot 8} + \frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12} + \cdots$$

I've been breaking my head over it for a very long time now. Any help would be greatly appreciated. Thank you.

Answer 1

You can simplify the equation to:$$1+\left(-\frac{3}{2}\right)\left(-\frac{1}{2}\right)+\frac{1}{2!}\left(-\frac{3}{2}\right)\left(-\frac{3}{2}-1\right)\left(-\frac{1}{2}\right)^2+\frac{1}{3!}\left(-\frac{3}{2}\right)\left(-\frac{3}{2}-1\right)\left(-\frac{3}{2}-2\right)\left(-\frac{1}{2}\right)^3+\cdots$$

By binomial expansion, we find that the above is equal to $(1-\frac{1}{2})^{-\frac{3}{2}}=\frac{1}{2}^{-\frac{3}{2}}=2^{\frac{3}{2}}=\sqrt{8}=2\sqrt{2}$.