Consider the sequence of sums (1)/(3), (1)/(3)+(1)/(9), (1)/(3)+(1)/(9)+(1)/(27) (1)/(3)+(1)/(9)

step 1 So this is an interesting kind of discussion. We have one -third, and then we just keep adding v

Consider the sequence of sums (1)/(3), (1)/(3)+(1)/(9),...

Consider the sequence of sums $\frac{1}{3}, \frac{1}{3}+\frac{1}{9}, \frac{1}{3}+\frac{1}{9}+\frac{1}{27}$ $\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}, \ldots$ (a) What happens to the terms of this sequence of sums as $k$ gets larger and larger? (b) Find a sufficiently large value of $k$ which will guarantee that every term past the $k$ th term of this sequence of sums is in the interval $(0.49999,0.5)$.

Consider the sequence of sums $\frac{1}{3}, \frac{1}{3}+\frac{1}{9}, \frac{1}{3}+\frac{1}{9}+\frac{1}{27}$
$\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}, \ldots$
(a) What happens to the terms of this sequence of sums as $k$ gets larger and larger?
(b) Find a sufficiently large value of $k$ which will guarantee that every term past the $k$ th term of this sequence of sums is in the interval $(0.49999,0.5)$.

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Key Concepts

Geometric Series

A geometric series is a sequence in which each term is obtained by multiplying the previous term by a fixed constant called the common ratio. This concept is fundamental in understanding how series behave, especially when determining whether they converge or diverge.

Sum of an Infinite Geometric Series

When the absolute value of the common ratio is less than one, an infinite geometric series converges to a finite limit given by the formula S = a/(1 - r), where a is the first term and r is the common ratio. This result helps explain the ultimate destination of the partial sums as more terms are added.

Partial Sum Formula

The sum of the first k terms of a geometric series is given by S_k = a(1 - r^k)/(1 - r). This expression is useful for quantifying how the series accumulates toward its limit and for estimating the error between the finite sum and the infinite sum.

Convergence and Error Estimation

Convergence refers to the behavior of a sequence of partial sums approaching a specific value as the number of terms increases. Error estimation involves determining how many terms are needed to ensure that the partial sum is within a desired tolerance of the limit. This concept is essential for answering questions about approximating infinite sums to a specified degree of accuracy.

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