Series DEFINITION: An infinite series is an expression that can be written in the form The numbers are called the terms of the series. Consider DEFINITION: Let be the sequence of partial sums of the series If the sequence converges to a limit then the series is said to converge to and is called the sum of the series. We denote this by writing If the sequence of partial sums diverges, then the series is said to diverge. A divergent series has no sum. Famous Series 1. 2. (Harmonic series) 3. 4. 5. 6.    7. Notation DEFINITION: EXAMPLES: (a)  (b)  (c)  (d)  (e)  (f)  (g)  (h)  (i)  (j)  EXAMPLE: Write using summation notation. Solution: We have EXAMPLE: Write using summation notation. Solution: We have EXAMPLE: Write using summation notation. Solution: We have EXAMPLE: Write using summation notation. Solution: We have EXAMPLE: Write using summation notation. Solution: We have EXAMPLE: Write using summation notation. Solution: We have EXAMPLE: Write using summation notation. Solution: We have Geometric Series THEOREM: A geometric series converges if and diverges if If the series converges, then the sum is Proof: We distinguish two cases: Case A: Assume that Then Case B: Assume that Then Subtracting these equations, we get Note that if then therefore Thus when the geometric series is convergent and its sum is If or the sequence is divergent and so, does not exist. Therefore, the geometric series diverges in those cases. EXAMPLE: Find the sum of the geometric series Solution: We have EXAMPLE: Find the sum of the geometric series Solution: We have EXAMPLE: Find the sum of the geometric series Solution: We have EXAMPLE: Find the sum of the geometric series Solution: We have EXAMPLE: Find the sum of the geometric series Solution: We have EXAMPLE: Find the sum of the geometric series Solution: We have PROBLEMS: (a) Is the series convergent or divergent? Solution: We have Since it follows that the series diverges. (b) Write as a ratio of two integers. Solution 1: Put then So therefore which gives Solution 2: We have (c) Find the sum Solution: We have (d) Find the sum Solution: We have Telescoping Sums EXAMPLE: Evaluate Solution: We have REMARK: In the same way one can prove that EXAMPLE: Find Solution: We have Therefore EXAMPLE: Show that for any positive integer Solution: We have Divergence Tests THEOREM (The Divergence Test): (a) If does not exist or if then the series diverges. (b) If then the series may either converge or diverge. THEOREM: If the series converges, then EXAMPLES: (a) diverges because does not exist. (b) diverges because (c) diverges because (d) diverges because (e) We have however diverges and converges. THEOREM: (a) If and are convergent series, then and are convergent series and the sums of these series are related by (b) If is a nonzero constant, then the series and both converge or both diverge. In the case of convergence, the sums are related by (c) Convergence or divergence is unaffected by deleting a finite number of terms from a series; in particular, for any positive integer the series both converge or both diverge. EXAMPLE: Find . Solution: We have EXAMPLE: Find . Solution: We have