Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

Previous problemNext problem

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

For which positive integers $k$ is the following series convergent? $\sum n_{1} \infty \frac{{(n!)}^{2}}{(k n)!}$

For all

k \geq 1

For all

k \geq 2

For all

k \geq 3

For no positive integer

k

Previous problemNext problem

Verified step by step guidance

Step 1: Begin by analyzing the given series: $ \sum_{n=1}^{\infty} \frac{(n!)^2}{(kn)!} $. The convergence of the series depends on the growth rate of the numerator $(n!)^2$ compared to the denominator $(kn)!$.

Step 2: Use the ratio test to determine convergence. The ratio test states that a series $ \sum a_n $ converges if $ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 $. Here, $ a_n = \frac{(n!)^2}{(kn)!} $.

Step 3: Compute $ \frac{a_{n+1}}{a_n} $: Substitute $ a_{n+1} = \frac{((n+1)!)^2}{((k(n+1))!)} $ and $ a_n = \frac{(n!)^2}{(kn)!} $ into the ratio $ \frac{a_{n+1}}{a_n} $. Simplify using factorial properties, such as $ (n+1)! = (n+1) \cdot n! $.

Step 4: Analyze the growth of $ (n!)^2 $ versus $ (kn)! $ as $ n \to \infty $. For small values of $ k $, the denominator $ (kn)! $ grows slower than $ (n!)^2 $, causing the series to diverge. For larger values of $ k $, $ (kn)! $ grows faster, leading to convergence.

Step 5: Conclude that the series converges for $ k \geq 3 $. For $ k = 1 $ or $ k = 2 $, the growth of $ (kn)! $ is insufficient to counterbalance $ (n!)^2 $, and the series diverges. For $ k \geq 3 $, the denominator grows sufficiently fast to ensure convergence.