Exercise · ZW-NUMERICAL-SERIES-008 · ★★☆ Intermediate
Compute \(\displaystyle\sum_{n=0}^{\infty}\left(-\tfrac23\right)^n\).
Choose one option.
Show answer
- B. \(\tfrac35\)
Explanation
The ratio is \(-\tfrac23\) and \(|r|<1\): the sum is \(\frac1{1+2/3}=\tfrac35\). The value \(\tfrac53\) comes from mishandling the sign of the ratio.
Review the concept: Numerical series
A series is the sequence of partial sums \(s_N\) of a sequence of terms \(a_n\); it converges if \(s_N\) has a finite limit, and that limit is its sum.
\[s_N=\sum_{n=1}^{N}a_n,\qquad \sum_{n=1}^{\infty}a_n=\lim_{N\to\infty}s_N\]
More practice problems: numerical series
- ZW-NUMERICAL-SERIES-001 Compute ∑_n=0^∞(tfrac12)^n.
- ZW-NUMERICAL-SERIES-002 Compute ∑_n=1^∞(1)/(n(n+1)).
- ZW-NUMERICAL-SERIES-003 For the series ∑_n=1^∞1n^2, compute the partial sum s_3.
- ZW-NUMERICAL-SERIES-004 What happens with ∑_n=0^∞(-1)^n?
- ZW-NUMERICAL-SERIES-005 Which of these series diverges because its general term does not tend to zero?
- ZW-NUMERICAL-SERIES-006 Compute ∑_n=1^∞(3)/(4^n).
- ZW-NUMERICAL-SERIES-007 Compute ∑_n=2^∞(1)/(n^2-1).
- ZW-NUMERICAL-SERIES-009 The partial sums of a series are s_N=(2N)/(N+1). What is the term a_3?