1. Prove that log(1+r) = geometric series. (-1)-1 for <1. The simplest way to do this...
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...
2. Find the Fourier Series of f(T). TER,(-2,2) (1) So, -2 <r<0, 2-I, 0<I<2.
1. Let x, a € R. Prove that if a <a, then -a < x <a.
Exercise 3. Suppose that |2 < 2. Prove that the series converges absolutely.
arn-1 is n=1 5. For positive numbers a and r, it is known that the geometric series > divergent. Which of the following is true? (a) r < -1. (b) -1<r<1. (c) r>1. (d) r > a.
Hint: use geometric series and the theorem on differentiation of a
power series
6.7 Obtain power series expansions for (1z <1. (Hint: use 6.11.) and for (1+z, each valid for l
4. Prove that SNS Here r < n and r < m.
5. If a, b E R, prove that abl < (a + b^).
3) Prove that there exists f : R → R non-negative and continuous such that f € L'OR : dm) ( i.e. SR \f|dm <00) and lim sup f(x) = ∞. 2-0
which way is the simplest way of solving this problem?
is there other ways to solve it without using gamma function? if
so, please show the different ways! thank you so much !
Let Yi, Y2, ..., Yso be a sequence of exponentially distributed random variable with parameter-4 , and Y =/i/40. Find P(Y<2) 40 /mail.google.com/mail/u/0/#inbox/FMcgxwBVCzcfwVpqNzGmrwWBgGqrN?projector-1 &messagePartid=0.2