What is the sum of the infinite geometric series #sum_(n=1)^oo6(0.9)^(n-1)# ?

Answer 1

This is a geometric series with first term 6 and common ratio 0.9 (see geometric series ). The common ratio is less than 1 in absolute value, so that the series is convergent and its sum is given by #"first term"/(1-"common ratio")=6/(1-0.9)=60#

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Answer 2

William Abdalla

The sum of the infinite geometric series ( \sum_{n=1}^{\infty} 6(0.9)^{n-1} ) can be calculated using the formula for the sum of an infinite geometric series, which is given by ( \frac{a}{1 - r} ), where ( a ) is the first term of the series and ( r ) is the common ratio.

In this series, the first term ( a = 6 ) and the common ratio ( r = 0.9 ).

Substituting these values into the formula, we get:

[ \frac{6}{1 - 0.9} = \frac{6}{0.1} = 60 ]

Therefore, the sum of the infinite geometric series is 60.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.