Differences

This shows you the differences between two versions of the page.

--- math121a-f23:hw_2 [2023/09/02 00:38]
pzhou [Calculus]
+++ math121a-f23:hw_2 [2026/02/21 14:41] (current)
@@ Line 1: / Line 1: @@
 ====== Homework 2 ======
+{{ :math121a-f23:math121a_hw2.pdf | solution}} (thanks to an anonymous student who provided the solution)
 I will update the homework after each lectures. It is due next Wednesday (since we have Labor day Monday)
 ===== Vector Space Problems =====
@@ Line 28: / Line 32: @@
   * $\sum_{n=1}^\infty n^2/n!$
-. Let $a_n$ be a sequence of $\pm 1$. Show that $\sum_{n=1}^\infty a_n / 2^n$ is convergent.
+. Let $a_n$ be a sequence of $\pm 1$. Show that $\sum_{n=1}^\infty a_n / 2^n$ is convergent. (Hint: absolute convergence implies convergence)
+. What is radius of convergence? Is it true that $$ \frac{1}{1-x} = 1+ x + x^2 + \cdots $$ holds for all real number $x \neq 1$?
+. We know that the following series diverge
+$$ 1 + 1/2 + 1/3+ 1/4 \cdots. $$
+Question: does the following alternating series converge? Why?
+$$ 1 - 1/2 + 1/3 - 1/4 + \cdots $$
+(Optional): Fix any real number $a$. Show that by rearrange the order of the terms in the above alternating series, we can have the series converges to $a$.
+. Line integral: let $\gamma$ be the straightline from $(0,0)$ to $(1,1)$. Compute the line integral
+$$ \int_\gamma 2 dx + 3 dy. $$
+What if we replace $\gamma$ by a curved line but still from $(0,0)$ to $(1,1)$, would the above result change? Why?
-(more coming on Friday night)

Lecture Notes