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Questions › DSE › DSE 2008 › Q55
2 marksMathematics › Sequences, series and convergence › Sequences and their limitsHardabout 4 min

DSE 2008 Question 55: Sequences, series and convergence

If A is a set of real numbers, let gA(x)=1g_{A}(x)=1gA​(x)=1 if x∈Ax\in Ax∈A and 000 if x∉Ax\notin Ax∈/A. Consider the sequence of functions fn(x)=n g[0,1/n](x)f_{n}(x)=n\,g_{[0,1/n]}(x)fn​(x)=ng[0,1/n]​(x) for all real xxx and each n=1,2,3,…n=1,2,3,\ldotsn=1,2,3,…. Then

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If A is a set of real numbers, let g_A(x)=1 if x∈ A and 0 if x∉ A. Consider the sequence of functions fₙ(x)=n g_([0,1/n])(x) for all real x and each n=1,2,3,…. Then (A) For every x, the sequence (fₙ(x))ₙ₌₁^∞ has a limit in the reals. (B) lim_(n→∞)fₙ(x) does not exist for any x. (C) When lim_(n→∞)fₙ(x) exists, the actual limit depends on the x in question. (D) lim_(n→∞)fₙ(x) exists for all but a finite set of real numbers x.

Also written as: If A is a set of real numbers, let g_A(x)=1 if x in A and 0 if x not in A. Consider the sequence of functions f_n(x)=n g_([0,1/n])(x) for all real x and each n=1,2,3,.... Then

Sequences, series and convergence has come up 33 times in 5 exams since 2006.

What this question tests

This question is on Sequences and their limits, part of Sequences, series and convergence in Mathematics.

Whether a sequence settles down and what it settles to.

TestStop has 10 past-paper questions on this subtopic.

Practise Sequences, series and convergenceMore on Sequences and their limits

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Question from the DSE 2008 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.