A geometric sequence has first term a and last terml and the sum of all these terms is

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.

5: Decide whether the series converges. If so, find its sum. \sum_{n=1}^{\infty} \frac{3}{n(n+1)} \sum_{n=2}^{\infty} \frac{3^{n}-1}{5^{n-2}} Use the Ratio or Root test, to determine whether the series converges or diverges. \sum_{n=1}^{\infty} \frac{(\ln n)^{2 n}}{n^{n+1}}

Let a, a, ag...., an... be an arithmetic sequence. Find a24 and S34- a₁ =4, d = 3

20.2.9 Prove that \frac{h}{2 \pi i} \int_{-\infty}^{\infty} \frac{e^{-i \omega t} d \omega}{E_{0}-i \Gamma / 2-\hbar \omega}=\left\{\begin{array}{ll} \exp \left(-\frac{\Gamma t}{2 \hbar}\right) \exp \left(-i \frac{E_{0} t}{\hbar}\right), & t>0 \\ 0, & t<0 \end{array}\right. This Fourier integral appears in a variety of problems in quantum mechanics: barrier penetration, scattering, time-dependent perturbation theory, and so on.

\text { Consider a sequence } f_{n}:[0, \pi] \rightarrow \mathbb{R} \text {, } f_{n}(x)=\left\{\begin{array}{ll} \sin (n x) & 0 \leq x \leq \frac{\pi}{n}, \\ 0 & \frac{\pi}{n} \leq x \leq \pi . \end{array}\right. ) Find pointwise limit of the sequence (fn) on [0, ]. Explain, why fn does not converge uniformly on1 [0, 1].