Suppose {an} and {bn} are sequences, and an = bn for all n ≥ 1000, but an ≠ bn for n < 1000. Is it true that lim n → ∞ an = lim n → ∞ bn? True False
This is a short reading exercise on reindexing. There is no standard name to this operation. Some people call it shifting the summation. Consider the notation for a series P∞ n=1 an. The symbol n is called the index of summation. The lower bound of the series is 1 . If we write P∞ n=u bn, then the lower bound of the series is u. Reindexing means to change lower bound of the series without changing the series. For example, the series 1 + 1 4 + 1 9 + 1 16 + 1 25 + • • • can be written as P∞ n=1 1 n2 or P∞ n=3 1 (n−2)2 . Please verify this by writing down the first few terms of both series. Therefore, we have an equality between series X∞ n=1 1 n2 = X∞ n=3 1 (n − 2)2 Notice we DO NOT simply replace n = 1 by n = 3. The expression for the terms of the series also have to change for the equality to hold. To reindex by symbolic manipulation, we introduce a change of variable. In this example, we start with the series P∞ n=1 1 n2 . We wish to reindex from n = 1 to n = 3, meaning the lower bound of the series changes from n = 1 to n = 3 without changing the series. We introduce m = n + 2 (because m = 3 when n = 1 ). Isolate for n to get n = m − 2, and we replace every occurrence of n in the series by the expression on the right-hand side (which is m − 2 in this example). The series becomes X∞ n=1 1 n2 = X∞ m−2=1 1 (m − 2)2 ( replace n by m − 2) = X∞ m=3 1 (m − 2)2 ( reorganize the lower bound of summation) = X∞ n=3 1 (n − 2)2 (change m back to n ) In general, if we wish to reindex a series P∞ n=u an from n = u to n = v, we introduce a change of variable m = n+(v−u) and perform similar manipulations as shown above. In the example above, u = 1 and v = 3. Let r be a non-zero constant. You are asked to reindex the series P∞ n=1 1 rn−1 from n = 1 to n = 0. What is the result of reindexing? Select one or more: a. P∞ n=0 1 rn+1 b. P∞ n=0 1 rn−1 c. P∞ n=0 1 rn
Suppose n/n+1 is the sequence of partial sums of the series ∑ n=1 ∞ an Determine the value of ∑ n=1 ∞ an.
Find a formula for the nth partial sum of the telescoping series below and use it to determine if the series converges or diverges. If the series converges, find its sum. ∑ n = 1 ∞ ( n + 2 − n + 1 ) A formula for the k t h term of the sequence of partial sums is Sk = . (Type an exact answer, using radicals as needed.) Evaluate lim k → ∞ Sk or state that the series diverges. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. lim k → ∞ Sk = (Type an integer or a fraction.) B. The series diverges.
Suppose {n n + 1} is the sequence of partial sums of the series ∑ n = 1 ∞ an. Determine the value of a3. Use a calculator to find a numerical approximation of your final answer to 2 decimal places. For example, if your final answer is π, then you should enter 3.14 in the box below. In tests and the final exam, you will NOT need to approximate your final answer numerically. Answer: