Question

Suppose (an)n≥1 and (bn)n≥1(a_n)_{n \geq 1} \ and\  (b_n)_{n \geq 1}(an​)n≥1​ and (bn​)n≥1​​ are two bounded sequences of real numbers.  Which of the following is true?

Accepted Answer

C

Answer

​lim sup⁡n→∞(an+(−1)nbn)=lim sup⁡n→∞an+∣lim sup⁡n→∞bn∣\limsup_{n 	o \infty} (a_n + (-1)^n b_n) = \limsup_{n 	o \infty} a_n + |\limsup_{n 	o \infty} b_n|n→∞limsup​(an​+(−1)nbn​)=n→∞limsup​an​+∣n→∞limsup​bn​∣​​

Answer

​lim sup⁡n→∞(an+(−1)nbn)≤lim sup⁡n→∞an+lim sup⁡n→∞bn\limsup_{n 	o \infty} (a_n + (-1)^n b_n) \leq \limsup_{n 	o \infty} a_n + \limsup_{n 	o \infty} b_nn→∞limsup​(an​+(−1)nbn​)≤n→∞limsup​an​+n→∞limsup​bn​​​

Answer

​lim sup⁡n→∞(an+(−1)nbn)≤lim sup⁡n→∞an+∣lim sup⁡n→∞bn∣+∣lim inf⁡n→∞bn∣\limsup_{n 	o \infty} (a_n + (-1)^n b_n) \leq \limsup_{n 	o \infty} a_n + |\limsup_{n 	o \infty} b_n| + |\liminf_{n 	o \infty} b_n|n→∞limsup​(an​+(−1)nbn​)≤n→∞limsup​an​+∣n→∞limsup​bn​∣+∣n→∞liminf​bn​∣​​

Answer

​lim sup⁡n→∞(an+(−1)nbn)\limsup_{n 	o \infty} (a_n + (-1)^n b_n)n→∞limsup​(an​+(−1)nbn​) may not exist.​

Suppose $(a_n)_{n \geq 1} \ and\ (b_n)_{n \geq 1}$ are two bounded sequences of real numbers.
Which of the following is true?

$\limsup_{n \to \infty} (a_n + (-1)^n b_n) = \limsup_{n \to \infty} a_n + |\limsup_{n \to \infty} b_n|$

$\limsup_{n \to \infty} (a_n + (-1)^n b_n) \leq \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n$

$\limsup_{n \to \infty} (a_n + (-1)^n b_n) \leq \limsup_{n \to \infty} a_n + |\limsup_{n \to \infty} b_n| + |\liminf_{n \to \infty} b_n|$

$\limsup_{n \to \infty} (a_n + (-1)^n b_n)$ may not exist.

Correct option is C

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Suppose (an)n≥1 and (bn)n≥1(a_n)_{n \geq 1} \ and\ (b_n)_{n \geq 1}(an​)n≥1​ and (bn​)n≥1​​ are two bounded sequences of real numbers. Which of the following is true?

​lim sup⁡n→∞(an+(−1)nbn)=lim sup⁡n→∞an+∣lim sup⁡n→∞bn∣\limsup_{n \to \infty} (a_n + (-1)^n b_n) = \limsup_{n \to \infty} a_n + |\limsup_{n \to \infty} b_n|n→∞limsup​(an​+(−1)nbn​)=n→∞limsup​an​+∣n→∞limsup​bn​∣​​

​lim sup⁡n→∞(an+(−1)nbn)≤lim sup⁡n→∞an+lim sup⁡n→∞bn\limsup_{n \to \infty} (a_n + (-1)^n b_n) \leq \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_nn→∞limsup​(an​+(−1)nbn​)≤n→∞limsup​an​+n→∞limsup​bn​​​

​lim sup⁡n→∞(an+(−1)nbn)\limsup_{n \to \infty} (a_n + (-1)^n b_n)n→∞limsup​(an​+(−1)nbn​) may not exist.​

Correct option is C

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Suppose $(a_n)_{n \geq 1} \ and\ (b_n)_{n \geq 1}$ are two bounded sequences of real numbers.
Which of the following is true?

$\limsup_{n \to \infty} (a_n + (-1)^n b_n) = \limsup_{n \to \infty} a_n + |\limsup_{n \to \infty} b_n|$

$\limsup_{n \to \infty} (a_n + (-1)^n b_n) \leq \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n$

$\limsup_{n \to \infty} (a_n + (-1)^n b_n)$ may not exist.