Show that the sequences is convergent with monotonic sequence theorem kg = 1 *3 2.4.6 b)...
4. Consider the sequence {z,.) such that z1-0, z2-1 and æn-telths,Yn (i) Show that (n) is convergent by showing that the subsequence of odd-indexed terms is monotonic increasing and subsequence of even-indexed terms is monotonic decreasing (ii) Find the limit of {%) (Hint: Consider x,-h-i) 4. Consider the sequence {z,.) such that z1-0, z2-1 and æn-telths,Yn (i) Show that (n) is convergent by showing that the subsequence of odd-indexed terms is monotonic increasing and subsequence of even-indexed terms is monotonic...
1. Show me what you know about monotonic sequences! a) What does it mean for a sequence to be monotonic? Clearly state the definition using full sentences. A good start is, "A sequence {an} is monotonic if and bnly if..." le b) Give an example of a sequence that is not monotonic. 10 iro c) Give an example of a monotonic sequence. d) Prove that your example sequence in Part C is in fact monotonic.
(a) Use Bounded Monotonic Sequence Theorem to show that the sequence with the given nth term converges, (b) Graph the first 10 terms of the sequence and estimate its limit. an
please do both question clearly,thank you! Consider the sequence of real numbers 13. 1 1 2' 1 2 + 2 + 1 2 + Show that this sequence is convergent and find its limit by first showing that the two sequences of alternate terms are monotonic and finding their limits. Prove that any sequence in hence we may suppose that each subsequence has a least term.) (Note that this result and the theorem on the convergence of bounded monotonic sequences...
4. Use the Monotone Convergent Theorem (Theorem 4.3.3) to prove that the following sequence is convergent, then find its limit. (Hint: You will need mathematical induction). S1 = 1 and Sn+1 = (2 sn + 5) forn EN
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and y1 = 8 and for n = 1,2,3,··· x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y . nn nn (a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all positive integers n. (xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive integers n. Hence, prove...
theorem1 let an and bn be squences of real numbers theorem 2 let an and bn and cn be squences of real numbers if an<bn<cn theorem 3 let an be squences of real numbers if an=L and L defined at all an,f(an)=f(L) theorem 4 f(x) defined for all x>n0 then limit f(x)=L and limit an =L theorem 5 follwing six squences converage to be limit limit lnn\n =0 ,limit (1+x/n)n=ex .... Based on Theorems 1 to 5 in Section 10.1...
6. Given an n21 2n+3) a) Is the sequence monotonic? b) Is the sequence bounded? c) Does the sequence converge or diverger
Suppose an- is a decreasing sequence of non-negative numbers (that is, 0 S an+1 S an for all n) a) Show that 2K a1 + - n-1 b) Suppose Σ-1 an is a convergent series. Use part a to show that Σ-1 2na2n converges. HINT: recall the monotonic sequence theorem c) Show that n-1 d) Suppose that Ση_1 2na2n is a convergent series. Use part c to show that Ση-1 an e shown that Ση.1 an conv Σ-1 2na2n converges....
z-1+ li and w-4-2, calculate zio, Izllwl and z20 3. a) [3 points lf b) 2 points Assume that ()n21 and (Jn)nz1 are two convergent sequences that limn→ㆀ-m-L1 and limn→ㆀyn-L2. Using the definition of the limit, show that the sequence (2xn-3yn + 5)n21 İs convergent and that limn-w(2x,-3yn + 5)-2L1-3L2 + 5. z-1+ li and w-4-2, calculate zio, Izllwl and z20 3. a) [3 points lf b) 2 points Assume that ()n21 and (Jn)nz1 are two convergent sequences that limn→ㆀ-m-L1...