Exam (elaborations)
Infinite Series solved questions
Infinite Series solved questions
[Show more]
Seller
Follow
Content preview
CHAPTER 37Infinite Series
37.1 Prove that, if E an converges, then an=Q.
Let Then
37.2 Show that the harmonic series diverges.
etc. There-
fore, Alternatively, by the integral test,
37.3 Does imply that E an converges?
No. The harmonic series E 1/n (Problem 37.2) is a counterexample.
37.4 Let Sn = a + ar + •• • + ar" ', with r^l. Show that
rS=ar + ar2 + - - - + ar" + ar". S. = a + ar + ar2 + • • • + ar"~\ Hence, (r- 1)5_ = ar" - a = a(r" - I)
Thus,
37.5 Let a T^ 0. Show that the infinite geometric series and diverges if
By Problem 37.4, if M < i , since r"-*0; if \r\>\,
JSJ—»+°°, since |r| -*+<». If r = l, the series is a + a + a H , which diverges since a¥=0. If
r = — 1, the series is a — a + a — a + • • • , which oscillates between a and 0.
37.6 Evaluate
By Problem 37.5, with
37.7 Evaluate
By Problem 37.5, with
37.8 Show that the infinite decimal 0.9999 • • • is equal to 1.
0.999 • • • by Problem 37.5, with
37.9 Evaluate the infinite repeating decimal d = 0.215626262
By Problem 37.5, with
Hence,
312
, INFINITE SERIES 313
37.10 Investigate the series
Hence, the partial sum
The series converges to 1. (The method used here is called "telescoping.")
37.11 Study the series
So
Thus, the series converges to
37.12 Find the sum of the series 4 — 1 + j — & + • • • .
This is a geometric series with ratio and first term a = 4. Hence, it converges to
37.13 Test the convergence of
This is a geometric series with ratio r = \ > 1. Hence, it is divergent.
37.14 Test the convergence of 3+I + I + I + - - - .
The series has the general term (starting with n = 0), but lim an = lim
Hence, by Problem 37.1, the series diverges.
37.15 Investigate the series
Rewrite the series as by Problems 37.11
and 37.10.
37.16 Test the convergence of
Hence, by Problem 37.1, the series diverges.
37.17 Study the series
So the partial sum
37.18 Study the series
Thus,
The partial sum
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller jureloqoo. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £6.47. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
83637 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy revision notes and other study material for 14 years now