Sequence and Series

Questions & Answers

The sum of the first n terms of a series if given by the expression 6-2n+1/3n-1. By finding an expression for the nth


Question 3. Assume La(x) is the line that is tangent to the graph of y = f(x) = x^2 at (a, f(a)). Let Aa be the point at which La(x) intersects the line x=0. (1) (1 point) Find the equation for La(x) for each a €R. (2) (1 point) Compute the distance between the points Ax and (x,f(x)). (3) (1 point) Find the rate of the change of the distance between A, and (x, f(x)) with respect to x. (4) (2 points) Use the definition of derivative to show that this rate is not defined at x= 0.


(1) Prove that the following sequence is a Cauchy sequence, s_{n}=1+\frac{2^{2}}{2 !}+\frac{3^{2}}{3 !}+\cdots+\frac{n^{2}}{n !}, \quad n \in \mathbb{N}


How long will it take money to double if it is invested at (A) 6% compounded continuously? (B) 8% compounded continuously? At 6% compounded continuously, the investment doubles in years.


Find the equation for any horizontal asymptotes for the function below. f(x)=\frac{x^{2}+2 x+5}{x-20} Find the horizontal asymptote(s). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.


A company is planning to manufacture snowboards. The fixed costs are $400 per day and total costs are $5200 per day at a daily output of 20 boards.


Q8


(3) Determine all values of p and q for which the following series converges: \sum_{k=2}^{\infty} \frac{1}{k^{a}(\ln k)^{p}}


(2) Test each of the following series for convergence: \text { (a) } \sum_{k=1}^{\infty} \frac{\sqrt{k+1}-\sqrt{k}}{k} \text { (b) } \sum_{k=1}^{\infty}(\sqrt[b]{k}-1)^{k} \text { (c) } \sum_{k=1}^{\infty} \sin \left(\frac{\ln k}{k^{2}}\right) \text { (d) } \sum_{k=1}^{\infty} \frac{1}{k} \ln \left(1+\frac{1}{k}\right)


1. Find explicit formulas for sequences of the form a₁, a2, a3, ... with the initial terms given below: 1/5, 3/20, 5/80, 7/320, 9/1280


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