11.2 Completed Notes 11.2: Series Definition: A series is the sum of the terms of a sequence. So, given a In calculating these partial sums, we actually created another sequence. sequence {an}, the series is a1 + a2 + ..., denoted an. Definition: The sequence of partial sums, sn, is defined by s1 = a1, Does it really make sense to talk about such things? After all, adding up s2 = a1 + a2, ..., sn = a1 + a2 + ... + an. sequences such as the positive integers would just be infinity. Definition: Let an be a series with partial sums sn = a1 + a2 + ... + an. If Consider the series by calculating "partial sums". the sequence of partial sums {sn} converges to some value s, i.e. sn = s, then we say an converges, and an = s. Otherwise, the series diverges. In other words, Definition: The geometric series is a + ar + ar2 + ... + arn ­ 1 + ... Example: Suppose a sequence a has partial sums s = . Does a n n n = arn ­ 1, for a ≠ 0. converge? Let's work out the convergence of the geometric series. First, we'll get an expression for sn. Remember that . With this in mind, the sum of the Theorem: The geometric series, arn ­ 1, converges if |r| < 1, and its sum is geometric series is: . If |r| ≥ 1, the geometric series diverges. Example: Determine if the following geometric series converge or diverge. If it converges, find its value. (a) 1 11.2 Completed Notes *(b) (d) 32n 2­n *(c) 4 + 6 + 9 + 13.5 + ... (e) *(f) Example: (Telescoping Series) Determine whether the series converges or diverges. If it converges, find its value. Example: (Telescoping Series) Determine whether the series Example: (Harmonic Series) Show that the series is divergent. converges or diverges. If it converges, find its value. (cont) 2 11.2 Completed Notes The previous theorem may not be very meaningful, but if we look at its Theorem: If the series an is convergent, then an = 0. contrapositive, we get a very useful test. Proof: The Divergence Test: If an ≠ 0 or does not exist, then an is divergent. IMPORTANT! If the limit is 0, this test gives us no information. The series may converge or diverge. For example, the harmonic series has a limit of 0 but diverges. A limit of 0 means that we know nothing! Example: Determine whether the following series are convergent or divergent. (a) (b) *(d) *(c) *(e) Theorem: If an and bn converge, then the following converge by these Example: Determine if the following series converge or diverge. If it formulas. converges, find its value. (1) (a) (2) (3) (b) Note: For rule (1). If the an series diverges, it also does. For rules (2) and (3), if one of the sums diverges, the overall sum diverges. Example: Determine if the following series converge or diverge. If it converges, find its value. (a) 3 11.2 Completed Notes More Telescoping Examples: Find the value of the following convergent *(c) telescoping series. (These questions will be mixed in with other series on Test 2 rather than telling you that they are telescoping.) (a) (a) (cont) (b) *(c) 4.