Calculus Volume 2 · 免费预览
无限级数
无限级数是对数列各项求和，其值是部分和数列的极限；等比级数与裂项级数可以精确求和，而调和级数则是典型的发散情形。
1. Partial sums and convergence
Partial sums and convergence
2. The geometric series
The geometric series
3. Telescoping series
Telescoping series
4. The harmonic divergence
The harmonic divergence
5. Series convergence definition
The infinite series sum of a_n converges when which condition holds?
6. Geometric sum
The geometric series sum from n = 0 to infinity of (1/2)^n equals what value?
7. Harmonic divergence
The harmonic series sum of 1/n diverges despite 1/n approaching zero. What does this demonstrate?
8. Telescoping collapse
The series sum of 1/(n(n+1)) telescopes to which value?
9. Nth term test limitation
The nth term test says a series whose terms do not approach zero diverges. What does it say about a series whose terms DO approach zero?
10. Geometric ratio condition
The geometric series converges when the absolute value of r is less than what number? Answer with a number.
11. Geometric first term
The geometric series with a = 3 and r = 1/2 sums to 3 over (1 - 1/2). What value results? Answer with a number.
12. Harmonic partial sum growth
The harmonic partial sums grow like ln N. As a single function name in lowercase, what function? Answer in lowercase.
13. Zeno resolution
Zeno's runner covers 1/2, then 1/4, then 1/8 of the remaining distance. The total covered is which geometric sum?
14. Telescoping boundary term
In the telescoping partial sum S_N = 1 - 1/(N+1), the vanishing term is 1 over what expression? Answer like N + 1.