For the following sequences, calculate their limits superior and inferior. (a) (b) ((-1)" (1-))1 (c) (n sin())1 (d) (0, 1, 2,0, 1, 3,0, 1, 4,...) (n²(-1+(-1)"))1
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- For each of the following sequences: (1) write the first five elements, (2) prove that the sequence converges and find its limit, and (3) determine the rate of convergence and the rate constantFind the lower sum for f(x)=x2 over [−1,1] with n=4. Give your answer in fractional form.The nth term of a sequence is represented by 2n4+25n2+32n−156n4+2n3−11n2−2n+17. What is the limit of the the nth term as x becomes increasingly large?
- Find the limit inferior and limit superior of the sequence {[5 sin n]}.Consider the sequence An , whose nth term is given by An = 2(1 + 0.025/n)^n where n is the number of interest payments per year on an initial investment of $2 made at an APR of 2.5%. Determine whether the sequence is convergent or divergent. If it is convergent, find the limit of the sequence and interpret the meaning of the limiting value.Find x so that x, x + 2, and x + 3 are consecutive terms of ageometric sequence.
- For ln(1+1/x), determine if the sequence is bounded and whether it is eventually monotone, increasing, or decreasing. How can you tell?Find the limit of the sequences with general term as given: (c) sin n2/ √n (d) n / n2 - 3(b) Give a qualitative explanation for why the sequence gn(x) = xn is not equicontinuous on [0, 1]. Is each gn uniformly continuous on [0, 1]?