{"id":10710,"date":"2017-09-08T12:29:59","date_gmt":"2017-09-08T19:29:59","guid":{"rendered":"https:\/\/magoosh.com\/hs\/?p=10710"},"modified":"2017-09-08T12:29:59","modified_gmt":"2017-09-08T19:29:59","slug":"ap-calculus-bc-review-series-convergence","status":"publish","type":"post","link":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-bc-review-series-convergence\/","title":{"rendered":"AP Calculus BC Review: Series Convergence"},"content":{"rendered":"

A series<\/strong> is the sum of infinitely many terms. But how can you add up an infinite<\/em> number of things? Well, it turns out that sometimes we can do exactly that! But first we’ll need to understand the concepts of series convergence and divergence.<\/p>\n

\"Series
Infinite series convergence may seem mysterious at first, but it’s not<\/em> the work of aliens…. so far as I know.<\/figcaption><\/figure>\n

Working with Series<\/h2>\n

Typically, a series is expressed either by writing out a few terms in order to establish a pattern, or by using sigma notation<\/strong>. <\/p>\n

\"Series<\/p>\n

Here, an<\/sub><\/em> is the general term<\/strong> for the series.<\/p>\n

Check out the following article for more explanation as well as examples: AP Calculus BC Review: Series Fundamentals<\/a>.<\/p>\n

Partial Sums<\/h3>\n

Any time that infinity is involved, we have to be extra careful with our definitions. So how do you add up infinitely many items? One item at a time! This is a job for partial sums<\/em>.<\/p>\n

A partial sum is something like a running total<\/em> for a series.<\/p>\n

Let   \"series\"   be a series.<\/p>\n

The k<\/em>th partial sum<\/strong> of the series is:<\/p>\n

\"Partial<\/p>\n

In other words, the k<\/em>th partial sum is the sum of the first k<\/em> terms of the series.<\/p>\n

The first four partial sums of a series are:<\/p>\n

\"first<\/p>\n

Now, the useful thing about partial sums is that they form a sequence<\/em>,<\/p>\n

\"sequence<\/p>\n

And then we define series convergence in terms of the convergence of this sequence of partial sums.<\/p>\n

Series Convergence and Divergence — Definitions<\/h3>\n

A series Σan<\/sub><\/em> converges<\/strong> to a sum S<\/em> if and only if the sequence of partial sums converges to S<\/em>. That is, a series converges if the following limit exists:<\/p>\n

\"Definition<\/p>\n

Otherwise, if the limit of sk<\/sub><\/em> (as k<\/em> → ∞) is infinite or fails to exist, then the series diverges<\/strong>.<\/p>\n

Ways That a Series Could Diverge<\/h3>\n

A series may diverge in three different ways:<\/p>\n

    \n
  1. The series diverges to infinity<\/strong> (∞) if the partial sums increase without bound. That is, sk<\/sub><\/em> → ∞.<\/li>\n
  2. The series diverges to negative infinity<\/strong> (-∞) if the partial sums decrease without bound. That is, sk<\/sub><\/em> → -∞.<\/li>\n
  3. If the limit of partial sums does not exists and is not ∞ or -∞, then we just say that the series diverges. This case occurs whenever the sequence of partial sums oscillates among multiple values or jumps around randomly.<\/li>\n<\/ol>\n

    Limit Test for Divergence<\/h2>\n

    There is a quick and easy test that can be used to show that a series diverges. (However, it can’t be used to show a series converges.)<\/p>\n

    Limit Divergence Test:<\/strong> If the limit of the general term of a series is not equal to 0, then the series diverges.<\/p>\n

    That is, if \"limit, then the series Σan<\/sub><\/em> diverges.<\/p>\n

    Unfortunately, if the limit does turn out to be zero, then the test is inconclusive<\/em>. Then you’d have to use additional convergence tests to figure out series convergence or divergence.<\/p>\n

    Let’s take a look at a few examples.<\/p>\n

    Using the Limit Divergence Test<\/h3>\n

    Which of the following series must diverge according to the Divergence Test?<\/p>\n

    \"four<\/p>\n

    Solutions<\/h4>\n

    Remember, for this question, we’re only allowed to use the Divergence Test.<\/p>\n

    (a) As n<\/em> → ∞, the values of 2n<\/sup><\/em> also approach ∞. The bottom line is that the limit of 2n<\/sup><\/em> is certainly not equal to 0.<\/p>\n

    Therefore, series (a) diverges.<\/p>\n

    (b) \"limit<\/p>\n

    Here, the answer is not clear. A limit of 0 does not automatically mean that the series will not diverge. So the Limit Divergence Test alone cannot say anything more about series convergence or divergence here.<\/p>\n

    In fact, we will prove that this series does converge by other methods.<\/p>\n

    (c) Again, the Divergence Test is inconclusive, because 1\/n<\/em> → 0 as n<\/em> → ∞.<\/p>\n

    Now this series actually diverges<\/em>. It’s called the Harmonic Series<\/strong>, and it’s just the sum of all of the reciprocals of the natural numbers.<\/p>\n

    \"harmonic<\/p>\n

    We’ll need another method to decide for sure that the Harmonic Series diverges. It’s far from obvious!<\/p>\n

    (d) \"limit<\/p>\n

    With a nonzero limit, the Divergence Test conclusively states that this series must diverge.<\/p>\n

    Geometric Series<\/h2>\n

    Let’s take a closer look at Examples (a) and (b). Both of these are geometric<\/strong> series. <\/p>\n

    A geometric series is the sum of the powers of a constant base. <\/p>\n

    \"Geometric<\/p>\n

    There is a straightforward test to decide whether any geometric series converges or diverges.<\/p>\n