# Power Series and Exact Values of Numerical Series

Power series and the exact values of numerical series are fundamental concepts in mathematics, particularly in calculus and analysis. A power series is an infinite series of the form ∑(aₙxⁿ), where each term is a polynomial function of x raised to a non-negative integer power. Understanding power series enables mathematicians to express functions as infinite sums, facilitating analysis and approximation techniques. Exact values of numerical series, on the other hand, involve determining precise sums of specific sequences, offering insights into various mathematical phenomena such as convergence, divergence, and the behavior of functions at different points.

Questions

- How do you use a power series to find the exact value of the sum of the series #1+e+e^2 +e^3 +e^4 + …# ?
- How do you use a power series to find the exact value of the sum of the series #1+2+4/(2!) +8/(3!) +16/(4!) + …# ?
- How do you use a power series to find the exact value of the sum of the series #1-pi^2/(2!)+pi^4/(4!) +pi^6/(6!) + …# ?
- How do you use a power series to find the exact value of the sum of the series #pi/4-(pi/4)^3/(3!)+(pi/4)^5/(5!)-(pi/4)^7/(7!) + …# ?
- How do you use a power series to find the exact value of the sum of the series #1-1/2+1/3-1/4 + …# ?
- How do you use a power series to find the exact value of the sum of the series #1+sqrt(2)+2/(2!) +(sqrt(2))^3/(3!) +4/(4!) + …# ?
- How do you use a power series to find the exact value of the sum of the series #1-(pi/4)^2/(2!)+(pi/4)^4/(4!) -(pi/4)^6/(6!) + …# ?
- How do you use a power series to find the exact value of the sum of the series #pi-pi^2/2+pi^4/(4!) -pi^6/(6!) + …# ?
- What is the relative maximum value for #f(x)=ln(x+5.1)-.5(x+3)^2+11.02#?