Workshop #10, Math 162 (Spring 2017) Week of April 3rd, 2017. Today’s Topics:

• Test for divergence • Telescoping series

• Geometric series test • The harmonic series

Warm-up question Consider the following sequences for n ≥ 0. Without calculating, predict the limits of each of the sequences as n → ∞. Explain what convergence or divergence of a sequence means.

3n − 1 n2 − 2n n3 − 3 1 a = a = a = a = n 2n + 2 n n3 + n + 1 n n2 + n + 1 n 2n + n2

1. Answer the following questions using the sequences defined in the warm-up.

(a) Sum the first three terms of each of the sequences. Are the terms up to a3 increasing or decreasing? How do you think this affects the long-term behavior of the series as you begin to add up infinitely many terms? ∞ X (b) Based on what you have discussed so far, guess the convergence behavior of the series an. (Don’t worry n=0 about justifying your guesses for now!) (c) What can you say about the relationship between conv/div of a sequence and conv/div of a series? ∞ 1 X (d) Provide your own example of a sequence an that converges to 2 such that an diverges. n=0 (e) Can the nth term test for divergence be used to show a series converges? (This is important!!)

∞ th X 2. Let Sk = k partial sum of the infinite series an. n=0 (a) First discuss what it means to “add up infinitely many things” with your partner(s). (b) What is the significance of the kth partial sum in determining the limiting behavior of an infinite series? k ∞ X X (c) Discuss and explain the following relationship: lim Sk = lim ai = ai k→∞ k→∞ i=1 i=1 (d) Now think of your own example of an infinite series that demonstrates the above relationship. Determine the lim Sk, and convergence or divergence of the series. k→∞

∞ X 1 3. Consider the series = 1 + 1 + ... n(n + 1) 1·2 2·3 n=1

1 (a) First determine the convergence/divergence of the sequence an = n(n+1) . (b) How else can the sequence be rewritten? (hint: think back to partial fractions)

(c) Using part (b), find the partial sums up to S5. What do you notice? (d) Can you write out the newly determined sequence of partial sums? th (e) Now find Sk, the k partial sum, and express it as a sum of just 2 terms.

(f) Having found this simplified form for Sk, determine whether the series converges or diverges. If it diverges, why? If it converges, find the value.

1 1 1 1 1 1 1 4. Having completed the last exercise, take a look at this series: (1 − 3 ) + ( 2 − 4 ) + ( 3 − 5 ) + ( 4 − 6 ) + ··· (a) Write the above series in sigma notation. (b) Before doing any calculations, do you think this series will converge or diverge? (c) Write out S5 and S6. (d) Now find the kth partial sum of the series. (e) Using part (c), determine whether the series is convergent/divergent. If it diverges, why? If it converges, what is the value? (f) Was your prediction in part (b) correct or not? Discuss. (g) Repeat the above steps for the series: (1 − 2) + (2 − 3) + (3 − 4) + ··· . Does anything change? (h) Provide a new example of a telescoping series whose sequence of partial sums converges to a value other than 1 or 1.5. No need to compute, by try to write down four or five examples of telescoping series.

3 3 5. Consider the following series: 6 − 3 + 2 − 4 + ... (a) Write the series in sigma notation, is it a geometric series? If so, identify the common ratio r. th (b) Discuss what the k partial sum of an infinite series, Sk is before continuing.

(c) Find the value of the partial sums of the series up to S5 and then plot a graph of (k, Sk). What can you predict about the convergence or divergence of the series based on the pattern observed? (d) Write the sigma notation for the finite sequence up to n = 7. Based upon conclusions in part (b) and (c), can you determine a "rule" for the summation of a finite geometric series?

(e) How is the sum of the infinite geometric series related to the sum of the finite series? Take the limit of Sk. (f) For an infinite geometric series, we are told that the series converges if the common ratio |r| is < 1. Explain and discuss why you think this is. (Hint: look back at your graph of {Sk}).

1 1 1 1 1 1 6. Now take a look at the following series: 1 + + + + + + + ... 3 2 9 4 27 8 (a) Is this series geometric? If so, identify the common ratio r, and determine if the series is convergent or divergent. (b) If it is divergent, explain why. (c) If it is convergent, provide the value of the sum.

∞ X 1 7. This exercise will help you understand why the harmonic series is divergent. n n=1

(a) Write out the terms of the series up to a8. What pattern is observed? (b) Discuss what it means for an infinite series to converge or diverge. What is expected to happen in each situation as you sum up infinitely many terms? th (c) Let S2k = k partial sum of the series. Find the value of the partial sums of the harmonic series up to S10 k and plot a graph of (2 ,S2k ). i. What pattern emerges as k increases?

ii. Try to write an inequality that represents this pattern for S2k

iii. Use your inequality to figure out how many terms would needed to be added in order to obtain S2k ≥ 20? (d) Having completed the above exercises, can you come up with a supported argument as to why the harmonic series diverges?

Review Problems

Determine whether each series is convergent/divergent. If divergent, why? If convergent, what is the value?

∞ ∞ n X X 3  1. [ln(n + 1) − ln(n)] 3. + n 4 n=1 n=1

∞ √ √ ∞ X n + 1 − n X 3n + 2n 2. √ √ 4. n + 1 · n 4n n=1 n=1