Prove that lim n→∞ n Σ(-1) k k k=1 exists and is finite.

Q: prove that if (xn) → 2, then (a) (2xn−1/3) → 1; (b) (1/xn) → 1/2. (For this exercise the Algebraic…

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Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108.

A: Note: As per our company guidelines we are supposed to answer the first question only. Kindly ask…

Q: Compute |4x – 12| lim x – 3

A: To evaluate:limx→3-|4x-12|x-3 Solution: we know, Modulus/absolute value function is defined as:…

Q: + bER satisfy O<b<1. Show

A: For  satisfying ,  using the binomail theorem.

Q: 1b. Show that lim inf (n) = = -∞ if and only if {n} is not bounded below.

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Q: Prove (sn) is bounded if and only if lim sup |Sn| < +∞.

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Q: M Let g(x):= 0 if x = [0, 1] is rational and g(x) = 1/x if x = [0, 1] is irrational. Explain why 8…

A: Given: gx:=0 and suppose x∈0,1 is rational and gx:=1x and suppose x∈0,1 is irrational. To find: Pn˙…

Q: Evaluate the limit n 1 2/3 lim 2 V3 3n n→∞ 1+ 3 + (k – 1)2v3' - k=1 3n

A: Given : limn→∞∑k=1n11+33+k-1233n2 233n Therefore, limn→∞∑k=1n11+33+k-1233n2…

Q: Compute lim n · (Vỹ – 1). | n→∞

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Q: i. Using Arithmetic of Limits, find limn→∞ 2n / (3n + 5)&nbsp. ii. Working directly from the…

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Q: Using Arithmetic of Limits, find limn→∞ 2n / (3n + 5)&nbsp. ii. Working directly from the definition…

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Q: Let (an) → 0, and use the Algebraic Limit Theorem to com- pute each of the following limits…

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Q: Suppose ƒ : E → R, p is a limit point of E, and lim f(x) exists. Prove that there exist x-p M> 0 and…

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Q: What is the limit lim x→1 X x² 1 1 + sin N|w 1 1 2 1-x Does not exist ∞ ?

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Q: 2. Assume that lim så and lim të exist, and tn ≥ sn for all n. Prove that limtë ≥ lim sɲ. (hint:…

A: Given that exist and for all n.The objective is to prove that .

Q: ) Suppose ƒ : E → R, p is a limit point of E C R, and lim f(x) > 0. Prove that there exist x-p c> 0…

A: To prove this statement, it can use the definition of a limit. The statement is essentially saying…

Q: Compute each of the following limits. Use exact values. 24 8t| lim t→ - 3 t + 3 |- 24 – 8t| lim t -…

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Q: lim sup (an + bn) lim inf a, + lim inf b, n→∞ n→∞ n→∞

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Q: 3. Show that if then lim an n→∞ = lim a3n+2 nx L = L =

A: We can write n = (3n+2) - 1 - 2n/3. As n approaches infinity, 2n/3 approaches infinity as well.…

Q: Prove that for every v = (x, y) E R? there exists a sequence v, = (xn, Yn) € R2, where rn, Yn are…

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Q: Let {z} be a sequence of complex numbers (z = x + ¡y, where x, y_€ R)and z € C. Give the 12 E- N…

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Q: Let an be recursively defined by a₁ n≥ 1. Find lim an. 84x = 3/4, an+1 = = n n+2 . a for

A: To find the limit of the sequence

Q: 1-e¬* lim x→0+ 2x2 -00 00 None of these O 1

A: We need to evaluate limx→0+1-e-x2x2 = 1-e-02(0)2 = 1-10= 00 Hence we will use L Hopital's rule It…

Q: 1-e-* lim x→0+ 2x2 -00 00 O None of these 1 || O O O

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Q: 2x2 ) 3x Evaluate the given limit of lim if it exists. 1 x +1 O 1 O limit does not exist 2

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Q: 4. Determine lim 1 2n), and prove it. n+1 n+ 2 Hint: use the Squeeze Theorem, modeling your work on…

A: To find- Determine lim1n+1 + 1n+2 + .... + 12n, and prove it.

Q: In(x + 1) – ln (2x) lim X→1 х2 — 1 II

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Q: (1) Find the limit In (n + 1) lim In n Work for (1). In n In In (n + 1) lim lim n 00 %3D In n In n…

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Q: Find a sequence (Pn) of characteristic functions with the property that the limit given by (t) =…

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Q: а.) lim X+00 Vx+1 9х+1 =? b.) lim [x – In(e* + e¬*)] =? x+00

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Q: show that If an > 0, n = 1, 2,. .... In+1 In+1 lim inf 14x <lim inf n ≤ lim sup Van ≤ lim sup nx In…

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Q: . Calculate Lim п? +п —п by using your calculus and algebra skills n→0

A: Definition and methods help to find the required limit of the given function. A value is said to be…