Summary
Summary The Integral Test | Calculus II Notes
- Course
- Institution
Highlights theorems and gives detailed and explained examples on the integral test for convergence
[Show more]
Seller
Follow
dazyskiies
Content preview
3.3.2 The Integral Testthe integral testis for a series an thatwe cannotevaluate
explicitly
we think the
of series as the union of rectangles with an representing the area of a rectangle ofwidth
one and heightan
visualize the terms of the harmonic series:
11
t
I (7
=
!!!!
itshows the area ofthe shaded columns. it is
bigger than the area under the curve y=b with 12x=5
·Ein S,dx
to continue to infinite the
ifthecolumswere
this
gives us an improper integral we can evaluate
-
(,* (logIx))1P = +
=
the areaunder the curve
diverges to to so the area represented by the columns also diverges to
00
+
example:Enz
Ein ), dx
Eines. Ende
1,*2dx *
=
-
- TE
it 1
=
in
+
the series must converge
Theorem:
letNo be
any natural number. If f(x) is a function which is define and continuous for all xx,No and which obeys
f(x) is
positive for all x =No
f(x) is a
decreasing function
f(n) On for all
= n= No
then Ian converges if (wif(x) dx converges
when the series converges, the truncation error
12an -Ean1= Sf(x) dx for all N =No
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 or Stuvia-credit 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 dazyskiies. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.49. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
52507 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now