Why does the geometric series with first term 1 and ratio $1/2$ converge to a finite limit despite having infinitely many terms?

Conditions

• First term $a=1$
• Common ratio $r=1/2$

Reasoning, step by step

1. Identify the parameters: first term is 1, ratio is $1/2$.
2. Observe the behavior of the partial sums $S_n$ as $n$ increases.
3. Apply the derived formula $S_n = 2(1 - 2^{-(n+1)})$.
4. Evaluate the limit as $n \to \infty$, where $2^{-(n+1)} \to 0$.
5. Conclude that the sum stabilizes at 2.

Example

The script states: 'Start with a geometric series whose first term is 1 and ratio is $1/2$. Its partial sums approach 2.' It further provides the explicit formula: '$S_n=2(1−2^{−(n+1)})$'.

Common misconceptions

• Believing that adding infinitely many numbers always results in infinity.
• Confusing the number of terms ($n+1$) with the index $n$ when calculating limits.

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Connected concepts

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• How does the convergence rate change if the ratio is closer to 1?
• What happens to the sum if the ratio is negative?