How do you calculate the nth partial sum $S_n$ for the geometric series starting at $k=0$ with ratio $1/2$ using the formula provided in the video?

Conditions

• Summation index $k$ runs from 0 to $n$
• Ratio $r = 1/2$

Reasoning, step by step

1. Confirm the indexing convention: sum from $k=0$ through $n$.
2. Recognize that this includes $n+1$ terms.
3. Use the specific formula given: $S_n = 2(1 - 2^{-(n+1)})$.
4. Substitute the value of $n$ into the equation.
5. Calculate the power $2^{-(n+1)}$ and subtract it from 1.
6. Multiply by 2 to get the final partial sum.

Example

The script explicitly defines the calculation method: 'Defining Sₙ as the sum from $k=0$ through n gives $n+1$ terms and Sₙ=$2(1-2^(-(n+1)))$. Keep the indexing convention consistent.'

Common misconceptions

• Using $2^{-n}$ instead of $2^{-(n+1)}$ due to incorrect counting of terms.
• Assuming the sum starts at $k=1$ rather than $k=0$.

Watch the explanation

Connected concepts

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• What is the difference between $S_n$ defined here and the sum of the first $n$ terms?
• How large must $n$ be for $S_n$ to be within 0.001 of 2?