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Which one of the following is FALSE for the sequence of functions {fn}, where fn(x)=1/1+nx, x below to (0,1), n=1,2,.? A. {fn} is convergent B. {fn} is uniformly convergent C. fn is continuous for all n D. fn>=fn+1, for all n.?
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Which one of the following is FALSE for the sequence of functions {fn}...
False statement for the sequence of functions {fn}

A. {fn} is convergent
- The sequence of functions {fn} is not convergent because as n approaches infinity, fn(x) approaches 0 for x in (0,1). This means that the limit of the sequence does not exist.

B. {fn} is uniformly convergent
- The sequence of functions {fn} is not uniformly convergent because for any ε > 0, there exists an x in (0,1) such that |fn(x) - 0| = |1/(1+nx) - 0| = 1/(1+nx) > ε for all n. This violates the definition of uniform convergence.

C. fn is continuous for all n
- Each function fn(x) = 1/(1+nx) is continuous on the interval (0,1) for all n, as it is a rational function with no points of discontinuity in this interval.

D. fn>=fn+1, for all n
- This statement is false because if we compare fn(x) and fn+1(x) for x in (0,1), we can see that fn(x) < fn+1(x)="" for="" all="" x.="" this="" is="" because="" the="" denominator="" (1+nx)="" is="" increasing="" as="" n="" increases,="" making="" fn+1(x)="" larger="" than="" fn(x)="" for="" all="" x="" in="" />
therefore, the false statement for the sequence of functions {fn} is option b: {fn} is uniformly convergent. fn+1(x)="" for="" all="" x.="" this="" is="" because="" the="" denominator="" (1+nx)="" is="" increasing="" as="" n="" increases,="" making="" fn+1(x)="" larger="" than="" fn(x)="" for="" all="" x="" in="" (0,1).="" therefore,="" the="" false="" statement="" for="" the="" sequence="" of="" functions="" {fn}="" is="" option="" b:="" {fn}="" is="" uniformly="">
therefore, the false statement for the sequence of functions {fn} is option b: {fn} is uniformly convergent.>
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Which one of the following is FALSE for the sequence of functions {fn}, where fn(x)=1/1+nx, x below to (0,1), n=1,2,.? A. {fn} is convergent B. {fn} is uniformly convergent C. fn is continuous for all n D. fn>=fn+1, for all n.?
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Which one of the following is FALSE for the sequence of functions {fn}, where fn(x)=1/1+nx, x below to (0,1), n=1,2,.? A. {fn} is convergent B. {fn} is uniformly convergent C. fn is continuous for all n D. fn>=fn+1, for all n.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Which one of the following is FALSE for the sequence of functions {fn}, where fn(x)=1/1+nx, x below to (0,1), n=1,2,.? A. {fn} is convergent B. {fn} is uniformly convergent C. fn is continuous for all n D. fn>=fn+1, for all n.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which one of the following is FALSE for the sequence of functions {fn}, where fn(x)=1/1+nx, x below to (0,1), n=1,2,.? A. {fn} is convergent B. {fn} is uniformly convergent C. fn is continuous for all n D. fn>=fn+1, for all n.?.
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