Exercise · ZW-NUMERICAL-SERIES-004 · ★☆☆ Basic
What happens with \(\displaystyle\sum_{n=0}^{\infty}(-1)^n\)?
Choose one option.
Show answer
- D. The series diverges: the partial sums alternate between \(1\) and \(0\)
Explanation
The partial sums are \(1,0,1,0,\dots\), which have no limit; by definition the series diverges.
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-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-008 Compute ∑_n=0^∞(-tfrac23)^n.
- ZW-NUMERICAL-SERIES-009 The partial sums of a series are s_N=(2N)/(N+1). What is the term a_3?