Answer:
(π+2)/(π-1)
Step-by-step explanation:
The sum of an infinite geometric series with first term 'a' and common ratio 'r' is given by the formula ...
S = a/(1 -r) . . . . . for |r| < 1
The given sum can be decomposed into a constant and a series:
= -2 +(3 +3/π +3/π² +3/π³ +...)
= -2 +S . . . where a=3 and r=1/π in the above sum formula
Then the sum is ...
[tex]-2+\dfrac{3}{1-\dfrac{1}{\pi}}=-2+\dfrac{3\pi}{\pi-1}=\dfrac{-2(\pi-1)+3\pi}{\pi-1}=\boxed{\dfrac{\pi+2}{\pi-1}}[/tex]
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Additional comment
There is no way to rationalize the denominator of this fraction, but the numerator can be rationalized by writing it as a mixed number:
[tex]=\dfrac{3}{\pi-1}+1[/tex]
Effectively, this is the same as we would have gotten with ...
1 +S . . . where a=3/π and r=1/π
