Which arrow best describes the first step in the mechanism for the reaction?

Which of the following series diverges by the Test for Divergence?a.   ∑n=1∞sinnπ2n+1 \sum_{n=1}^{\infty} \sin \left( \frac{n \pi}{ 2n+1} \right) b.   ∑n=1∞ 1en+1  \sum_{n=1}^{\infty} \frac{1}{ e^{n} +1}  c.  ∑n=1∞arctannn+1\sum_{n=1}^{\infty} \arctan \left( \frac{n}{n+1} \right)

Consider the region R bounded by   y = 4 x – x 2 y=4x-x^2 and y = 0 y=0 Find the volume obtained by rotation R about x = – 4 x =-4 Screenshot 2025-07-04 at 3.31.32 PM.png

If limn→∞ an  \lim_{n \to \infty} a_{n}  converges, then  limn→∞ a2n  \lim_{n \to \infty} a_{2n}  converges,

Use the graph of  y = tan x y =\tan x to answer the following questions   ∫ tan x   d x = ln | sec x | + C \left(\int \tan x \: dx = \ln |\sec x|+C \right) Screenshot 2025-07-04 at 2.57.27 PM.png a. Find area between the curve y = tan x y =\tan x and y = 0 y =0 from x = – π 3 x = -\frac{\pi}{3} to x = π 3 x = \frac{\pi}{3} b.  Consider the region R bounded by y = tan x y =\tan x , y = 0 y =0 and x = – π 3 x = -\frac{\pi}{3} .  Find the volume obtained by rotating R about the x-axis.

If  Sn=3n+42n+2 S_n = \frac{3n+4}{2n+2} is the partial sum of S=∑n=1∞ an S = \sum_{n=1}^{\infty} a_n ,find S, S ,.  a4 a_4

Assume that the sequence {an a_n } is increasing and bounded for all positive n. Find the limit of  an a_n a1=2 a_1 = 2 .   an+1=5-4an a_{n+1} = 5 -\frac{4}{a_n}

If ∑n=1∞an \sum_{n=1}^{\infty} a_n diverges, limn→∞  an=∞ \lim_{n \to \infty}  a_{n} = \infty

Compute the sum of the series∑n=2∞1n-1n+1 \sum_{n=2}^{\infty} \left(\frac{1}{\sqrt n} – \frac{1}{\sqrt {n+1}} \right)

If a square matrix A is invertible, which of the following must be true?