Prove that 1-[p(1-p)^0+p(1-p)^1+p(1-p)^2+p(1-p)^3...+p(1-p)^n]=(1-p)^n using geometric series equation.
Prove that 1-[p(1-p)^0+p(1-p)^1+p(1-p)^2+p(1-p)^3...+p(1-p)^n]=(1-p)^n using geometric series equation.
Find the sum of each geometric series: ΣXe) n +3 Σ-5 b) 50 n-0 n-1 Find the sum of each geometric series: ΣXe) n +3 Σ-5 b) 50 n-0 n-1
n-1 1 1 1 4. The series $0 = 1 + +... is a geometric series. 2 2 4 8 n=1 Which of the following is true? (a) The series is convergent and its sum is less than 1/2. (b) The series is convergent and its sum is 1/2. (c) The series is convergent and its sum is 2/3. (d) The series is convergent and its sum is more than 2/3.
(b) Use the identities above and the formula for the sum of a geometric series to prove that if n is an integer and je[1,2,..., n) then sin2 (2ntj/n) = n/2 t-1 so long as jメ1n/21, where Ir] is the greatest integer that is smaller than or equal to x. We were unable to transcribe this image (b) Use the identities above and the formula for the sum of a geometric series to prove that if n is an integer...
.... This is called the geometric series. 1. (a) Prove that 1+r+p2 + ... + -1 = (b) Use Riemann sums to calculate Pedro (Hint: At some point your Riemann sum may contain 1+e2/n + en + ... + 2(n-1)/n. What do you get if you set r = e2/n? You will probably have to use L'Hôpital's Rule at some point.)
Problem 1 Geometric Series. We will need to sum the geometric series to simplify some of the partition functions developed in class. Prove that the geometric series 7:0 for r| < 1. You may find it helpful to consider the partial sums Sj ?, xk 1+1+-.+4 and rSi =x+x2 + +z?+1 take the limit J ? 00, Can you see why the geomet ric series converges for r < 1 and diverges for ll 2 1 Explain. . You will...
12-1 + + 4. The series £9) .. is a geometric series. 4 n=1 Which of the following is true? (a) The series is convergent and its sum is less than 1/2. (b) The series is convergent and its sum is 1/2. (c) The series is convergent and its sum is 2/3. (d) The series is convergent and its sum is more than 2/3. IS 5. For positive numbers a and r, it is known that the geometric series divergent....
1. Prove that log(1+r) = geometric series. (-1)-1 for <1. The simplest way to do this is to integrate the
1. Express the sum m-1 k-0 in closed form. [Hint: The sum is a finite geometric series.] 2. Find the equation of the line connecting the endpoints of the graph of sin(x) on the interval [0, π/2] 1. Express the sum m-1 k-0 in closed form. [Hint: The sum is a finite geometric series.] 2. Find the equation of the line connecting the endpoints of the graph of sin(x) on the interval [0, π/2]
For natural number n, an = 1+1+3+--+--log n . x dt By use log x = hen x > 0,· w 1 t Prove that the series is convergence and for any n 2 1, - 0<an n+12n(n+1) For natural number n, an = 1+1+3+--+--log n . x dt By use log x = hen x > 0,· w 1 t Prove that the series is convergence and for any n 2 1, - 0
3. The following series is attributed to Newton. It can be used to calculate r. n-0 (n!) (2n 1) 24n+1 (a) (2 points) Prove that the series converges. (b) (2 points) Compare S5 to the actual value of π. 3. The following series is attributed to Newton. It can be used to calculate r. n-0 (n!) (2n 1) 24n+1 (a) (2 points) Prove that the series converges. (b) (2 points) Compare S5 to the actual value of π.