Find the function represented by the power series : f(x)=∑n=0∞ (-1)n2nxn+2n!f(x) = \sum_{n=0}^{\infty} \, \frac{ (-1)^{n}2^nx^{n+2}}{n!}

Determine if the series converges or diverges. Name the test you use for the following series. a. ∑n=1∞ (-4)n+132n \sum_{n=1}^{\infty} \frac{(-4)^{n+1}}{ 3^{2n} } b. ∑n=0∞cosnπn+1 \sum_{n=0}^{\infty} \cos \left( \frac{n \pi}{ n+1} \right) c. ∑n=1∞ lnnn \sum_{n=1}^{\infty} \: \frac{\ln n}{n}

Which series should be compared with ∑n=1∞ 2sinn+3n3/2 \sum_{n=1}^{\infty} \, \frac{2 \sin n +3}{n^{3/2} }

Find the 15th derivative of f(x) f(x) at x=2 x =2 for f(x)=∑n=0∞ (-1)n(x-2)n5n+1 f(x) = \sum_{n=0}^{\infty} \: \frac{(-1)^n (x-2)^n}{ 5^{n+1}}

Find the Maclaurin polynomial T4(x)T_4 (x) for f(x)=ln(1-x2) f(x) = \ln (1-x^2)

Use the geometric series 11-x=∑n=1∞ xn \frac{1}{1-x} =\sum_{n=1}^{\infty} \: x^n and you should not use the table of known Taylor or Maclaurin series for this problem. you should write the answer as the sigma notationa. (3 points) Write the Maclaurin series representation for the function: f(x)=11+x2 f(x) =\frac{1}{1+x^2} b. (4 points) From the expression you found in part (a), derive the Maclaurin series for h(x)=tan-1x h(x) = \tan^{-1} x

Use the graph and answer the following questions. a. (2 points) Find the area of R 1 R_1 b. (4 points) Find the volume of the solid generated by R 1 R_1 over the x-axis c. (4 points) Find the volume of the solid generated by R 2 R_2 over x = – 1 x =-1 Screenshot 2025-07-18 at 12.35.57 PM.png

A 80-kg  42-year-old male patient has myasthenia gravis. Bedside spirometry has found his VC to be 400 mL and NIF –19 cm H2O. Vital signs show a respiratory rate of 27 breaths/min, a heart rate of 134/minute, and blood pressure of 178/100 mm Hg. What should the respiratory therapist recommend?  

 Use the following data to calculate the P/F ratio ABG pH 7.19       PaCO2 82       PaO2 45        HCO3 22 Vent settings: AC PC   VT 385   RR 26   FiO2 100    PEEP 18

What is the force that results in the motion of electrons due to a difference in potential?