Calculate $\frac13+\frac29+\frac{1}{27}+\frac{2}{81}+\frac{1}{243}+\frac{2}{729}+\ldots$

sequences-and-series

How can we calculate the formula for the following series?

$\frac13+\frac29+\frac{1}{27}+\frac{2}{81}+\frac{1}{243}+\frac{2}{729}+\ldots$

I'm not quite sure what type of series it is anyway. It doesn't look geometric. Is there any strategy/procedure to follow in order to calculate the formula for a series of numbers?


Solution 1:

If you are looking for the infinite sum, it is the sum of two geometric series: $$ \sum_{k=1}^\infty\frac1{3^k}+\sum_{k=1}^\infty\frac1{9^k}=\frac12+\frac18=\frac58 $$

Solution 2:

If we add neighbouring pairs, we get $$\frac{5}{9}+\frac{5}{81}+\frac{5}{729}+\cdots.$$ This is an infinite geometric series with first term $\frac{5}{9}$ and common ratio $\frac{1}{9}$.

By the usual formula for the sum of an infinite geometric progression, the sum is $$\frac{5}{9}\frac{1}{1-\frac{1}{9}}.$$ This can be simplified to $\frac{5}{8}$.

Related

How to construct polynomial ring $K[x]$ over commutative ring $K$ by making use of universal arrows.
Show that $A = \{p\in\mathbb{ Q}^+\mid p^2<2\}$ contains no largest number and $B= \{p\in\mathbb {Q}^+\mid p^2>2\}$ contains no smallest number [duplicate]
How can I prove analytically the number $2^{100000}+1$ is not prime??
Sum of compact sets
Function grows slower than $\ln(x)$
If a matrix is triangular, is there a quicker way to tell if it is can be diagonalized?
Prove that identity element is unique
Is it possible to swap sums like that?
Matrix multiplication notation
Which set is greater in cardinality?
Finding the last two digits of a number
Why is this inference invalid?