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How do you find the n-th partial sum of a geometric series?

Answer 1

The formula to find the ( n )-th partial sum (( S_n )) of a geometric series is:

[ S_n = a \frac{1 - r^n}{1 - r} ]

Where:

( a ) is the first term of the series.
( r ) is the common ratio of the series.
( n ) is the number of terms in the partial sum.

This formula works for both finite and infinite geometric series, but for an infinite series, it's important to note that the sum only exists if ( |r| < 1 ). If ( |r| \geq 1 ), the series diverges.

If you're given the first term (( a )) and the common ratio (( r )), you can simply plug them into the formula to find the ( n )-th partial sum.

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Answer 2

Ezra Adams

Let us find a formula for the nth partial sum of a geometric series.

#S_n=a+ar+ar^2+cdots+ar^{n-1}#

by multiplying by #r#,

#Rightarrow rS_n=ar+ar^2+cdots+ar^{n-1}+ar^n#

by subtracting #rS_n# from #S_n#,

#Rightarrow (1-r)S_n=a-ar^n=a(1-r^n)#

(Notice that all intermediate terms are cancelled out.)

by dividing by #(1-r)#,

#Rightarrow S_n={a(1-r^n)}/{1-r}#

I hope that this was helpful.

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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.