So the power series [tex]\sum_{n=0}^{\infty}a_n (x-4)^n[/tex] converges at 0 and diverges at 9. How do we know that the limit of the sequence [tex]a_n[/tex] is zero? Thanks.
So the power series [tex]\sum_{n=0}^{\infty}a_n (x-4)^n[/tex] converges at 0 and diverges at 9. How do we know that the limit of the sequence [tex]a_n[/tex] is zero? Thanks.
Last edited by mmm4444bot; 12-02-2017 at 09:43 PM. Reason: Replaced $ and $ tags with tex and /tex tags
Last edited by mmm4444bot; 12-05-2017 at 06:46 PM. Reason: Replaced incorrect LaTex tags
Last edited by mmm4444bot; 12-05-2017 at 06:47 PM. Reason: Replaced incorrect LaTex tags
Well, first off I think there was a typo there and you meant to say [tex]\displaystyle \lim_{n \to \infty} a_n = 0[/tex]. Assuming that's the case, let's begin by reformulating the terms slightly. Let [tex]b_n = a_n \cdot (-4)^n[/tex] such that the series becomes:
[tex]b_0 + b_1 + b_2 + b_3 + b_4 + ...[/tex]
Now we've already established that this series converges, so what must be true about [tex]\displaystyle \lim_{n \to \infty} b_n[/tex]? Recall back to when you were first learning about series and you learned all those convergence tests. In particular, I'm thinking of the Divergence Test. From the results of this test, what can you then say about [tex]\displaystyle \lim_{n \to \infty} a_n[/tex]?
My understanding is that the divergence test only tells you that if the limit of the sequence [tex]a_n[/tex] is not zero, then the series diverges, and that it doesn't tell us that given a series that converges, its sequence must equal 0, which means that knowing that the series converges does not tell me that the sequence [tex]a_n[/tex] converges, is this correct?
Last edited by mmm4444bot; 12-05-2017 at 06:48 PM. Reason: Replaced incorrect LaTex tags
Last edited by mmm4444bot; 12-05-2017 at 06:49 PM. Reason: Replaced incorrect LaTex tags in quotation
"Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.
Actually, having re-read that link to Paul's Online Maths Notes, it is indeed that a convergent series has a sequence that tends to 0. I can see the solution to the original problem now. However, isn't the power series, rather than the sequence [tex]a_n[/tex], that converges, and so wouldn't the theory only apply to the power series, and so we still can't conclude anything about [tex]a_n[/tex] itself?
Last edited by mmm4444bot; 12-05-2017 at 06:55 PM. Reason: Replaced incorrect LaTex tags
If we take your original words at face value, "converges at zero", we can say something of the [tex]a_{n}[/tex], and we just did.
The first requirement of a convergent series is the convergence of the individual terms to zero. If that doesn't happen, the sequence of partial sums cannot converge.
More specifically, in this case, we know the series converges at x = 0. The sequence of terms has two pieces 1) [tex]a_{n}[/tex] and 2) increasing multiples of 4. Thus [tex]a_{n} \cdot 4^{n}[/tex] MUST find its way to zero (0). We know [tex]4^{n}[/tex] doesn't do that. What can we conclude about the [tex]a_{n}[/tex]?
Last edited by mmm4444bot; 12-05-2017 at 06:50 PM. Reason: Replaced incorrect LaTex tags in quotation
"Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.
@tkhunny great, I've got it now, thanks for clearing that up. It feels good to get that annoying bit of confusion out of the way.
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