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One of my homework problems asks, "Are the terms of the sequence n/(n+1) also getting closer and closer to π?" I'm confused because I thought that the limit was 1... Please help!

hot_queen
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Cynthia Galvan
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    The terms are indeed getting closer and closer to $\pi$. The point of the question is that we must be careful in defining $\lim_{n\to \infty}a_n=b$. We cannot simply say that it means that $a_n$ gets closer and closer to $b$. Another example is the sequence $1,\frac{1}{10},\frac{1}{2},\frac{1}{10^2},\frac{1}{3},\frac{1}{10^3}, \frac{1}{4}, \dots$. The terms are not getting "closer and closer" to $0$, sometimes they are very close, and later not very close, but the limit is $0$. And you are right, if $a_n=\frac{n}{n+1}$ the limit is $1$. – André Nicolas Jan 18 '14 at 02:50
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    It's also getting closer to 1000. – Bill Kleinhans Jan 18 '14 at 02:54
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    The terms are indeed 'getting closer and closer to $\pi$', but they never reach it (if that's what you want o know because the sequence is upper-bounded by $1$ – Alex Jan 18 '14 at 02:54
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    Technically, it is getting closer to $\pi$ since it's getting bigger and it is less than $\pi$. It's kind of like you're getting closer to the moon or sun as you reach for something on a shelf, though you know you'll never actually reach no matter how far you try to stretch your arms. – Brian Jan 18 '14 at 03:01
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    +1 LOL, yes, the book is right, I would have said no, it is getting closer to 1. But it also is getting closer to $e,pi$ or any number greater than 1 for that matter. Very simple trick question to confuse the hell outta of people. – jimjim Jan 18 '14 at 03:53

2 Answers2

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I've decided to post my comment as an answer.

Technically, it is getting closer to π since it's getting bigger and it is less than π. It's kind of like you're getting closer to the moon or sun as you reach for something on a shelf, though you know you'll never actually reach no matter how far you try to stretch your arms.

Brian
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Since the limit is $1$ the terms aren't getting close to $\pi$. I think the question wanted to check if you know the fact that if a sequence converges to a number, then the terms can't get close to a different number; in other words, the limit is unique, if it exists.

voldemort
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