Find the sum of the following series to infinity : 8 + 4√2 + 4 + ..

1.	Find the sum of the following series to infinity : 8 + 4√2 + 4 + ...... ∞
Answer» We observe that the above progression possess a common ratio. So it is a geometric progression. Common ratio = r = \(\frac{4\sqrt{2}}{8} = \frac{1}{\sqrt{2}}\) Sum of infinite GP = \(\frac{a}{1-r}\) ,where a is the first term and r is the common ratio. Note: We can only use the above formula if \|r\|<1 Clearly, a = 8 and r = \(\frac{1}{\sqrt{2}}\) ⇒ sum = \(\cfrac{8}{1-\frac{1}{\sqrt{2}}}\) = \(\cfrac{8\sqrt{2}}{\sqrt{2}-1}\)

Find the sum of the following series to infinity : 8 + 4√2 + 4 + ...... ∞

Answer»

We observe that the above progression possess a common ratio. So it is a geometric progression.

Common ratio = r = \(\frac{4\sqrt{2}}{8} = \frac{1}{\sqrt{2}}\)

Sum of infinite GP = \(\frac{a}{1-r}\) ,where a is the first term and r is the common ratio.

Note: We can only use the above formula if |r|<1

Clearly, a = 8 and r = \(\frac{1}{\sqrt{2}}\)

⇒ sum = \(\cfrac{8}{1-\frac{1}{\sqrt{2}}}\) = \(\cfrac{8\sqrt{2}}{\sqrt{2}-1}\)