Problem 2 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Summation notation review (a) Write the following in summation notation. i. \(3-4+5-6+7-\cdots-300\) ii. \(2+4+6+\cdots+1000\) iii. \(1+3+5+\cdots+999\) iv. \(\frac{2}{3}-\frac{2}{9}+\frac{2}{27}-\frac{2}{81}+\cdots+\frac{2}{3^{15}}\) v. \(x+x^{2}+x^{3}+x^{4}+\cdots+x^{40}\) vi. \(1^{2}+2^{2}+3^{2}+\cdots+100^{2}\) vii. \(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\cdots+a_{n} x^{n}\) (b) Write out the following sums. i. \(\sum_{i=2}^{5} i^{2}\) ii. \(\sum_{k=0}^{4} 2^{k}\) iii. \(\sum_{j=0}^{3} a_{j} x^{j}\)

Step-by-Step Solution

Answer

The series in summation notation would be: i. \( \sum_ {k=2}^{150} (-1)^k(2k+1) \) ii. \( \sum_ {k=1}^{500} 2k \) iii. \( \sum_ {k=0}^{499} (2k+1) \) iv. \( \sum_ {k=1}^{15} \frac{(-1)^{k+1}2}{3^k} \) v. \( \sum_ {k=1}^{40} x^k \) vi. \( \sum_ {k=1}^{100} k^2 \) vii. \( \sum_ {k=0}^{n} a_kx^k \) The following sums are written out as: i. 54 ii. 31 iii. \( a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3} \)

Step 1: Write series in summation notation

Identify the pattern, the starting and ending terms, and the increment for each series. Write these series in the compact form of summation notation. For example, i. \(3-4+5-6+7-\cdots-300 = \sum_ {k=2}^{150} (-1)^k(2k+1)\) ii. \(2+4+6+\cdots+1000 = \sum_ {k=1}^{500} 2k\) iii. \(1+3+5+\cdots+999 = \sum_ {k=0}^{499} (2k+1)\) iv. \(\frac{2}{3}-\frac{2}{9}+\frac{2}{27}-\frac{2}{81}+\cdots+\frac{2}{3^{15}} = \sum_ {k=1}^{15} \frac{(-1)^{k+1}2}{3^k}\) v. \(x+x^{2}+x^{3}+x^{4}+\cdots+x^{40} = \sum_ {k=1}^{40} x^k\) vi. \(1^{2}+2^{2}+3^{2}+\cdots+100^{2} = \sum_ {k=1}^{100} k^2\) vii. \(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\cdots+a_{n} x^{n} = \sum_ {k=0}^{n} a_kx^k\)

Step 2: Write out the sums

Replace the summation index by each of its values and perform the indicated operations to write out the sums. For example,i. \(\sum_{i=2}^{5} i^{2} = 2^{2}+3^{2}+4^{2}+5^{2} = 4+9+16+25 = 54\)ii. \(\sum_{k=0}^{4} 2^{k} = 2^{0}+2^{1}+2^{2}+2^{3}+2^{4} = 1+2+4+8+16 = 31\)iii. \(\sum_{j=0}^{3} a_{j} x^{j} = a_{0} x^{0}+a_{1} x^{1}+a_{2} x^{2}+a_{3} x^{3} = a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\)

Key Concepts

Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence, where each term after the first is obtained by adding a constant difference to the previous term. To identify this kind of series, look for a sequence of numbers with a uniform increase or decrease.

For example, the series given as 2+4+6+...+1000 is an arithmetic series where each term increases by 2. In summation notation, it can be written as \(\sum_{k=1}^{500} 2k\) which represents the sum of all terms from the first (when k = 1) to the 500th term (when k = 500).

To find the sum of an arithmetic series, we can use the formula \(S_n = \frac{n}{2}(a_1 + a_n)\), where S_n is the sum of the series, n is the number of terms, a_1 is the first term, and a_n is the last term. This approach simplifies the process and avoids the need to add up every single term individually.

Geometric Series

A geometric series is the sum of the terms of a geometric sequence, where each term is found by multiplying the previous one by a constant ratio. You can spot a geometric series if each term changes by a constant factor instead of a constant sum.

The sequence given as \(\frac{2}{3}-\frac{2}{9}+\frac{2}{27}-...+\frac{2}{3^{15}}\) forms a geometric series. It can be written in summation notation as \(\sum_{k=1}^{15} \frac{(-1)^{k+1}2}{3^k}\), with a common ratio of \(\frac{1}{3}\) between terms.

To calculate the sum of a finite geometric series, one can use the formula \(S_n = a_1\frac{1-r^n}{1-r}\), where a_1 is the first term, r is the common ratio, and n is the number of terms. For an infinite geometric series where |r| < 1, the sum converges to \(S = \frac{a_1}{1-r}\). This is particularly useful since it gives us a way to find the sum of an infinitely long series with diminishing terms.

Infinite Series

An infinite series continues indefinitely, without a final term. When such a series has a constant ratio between successive terms, and this ratio is between -1 and 1 (not inclusive), the series is a convergent geometric series and we can calculate its sum.

As seen earlier in the geometric series section, the infinite series \(x+x^{2}+x^{3}+...\) can be summed up if |x| < 1 using the formula for a geometric series. For infinite series that aren't geometric or don't have a clear pattern, convergence tests are used to determine if the series has a finite sum.

An important aspect of infinite series is understanding when they converge or diverge. Infinite series that do not converge are said to diverge, which means they do not sum up to a finite limit. Understanding and identifying convergence is crucial in analyzing infinite series.

Sequences and Series

The terms sequences and series may often be used interchangeably, but they have distinct meanings in mathematics. A sequence is an ordered list of numbers, and a series is the summation of the terms of a sequence. In simple terms, a sequence is like a list while a series is like a total.

A sequence can be anything from random to highly regulated in its order of numbers, while a series is about adding those numbers up. For example, the summation notation \(\sum_{k=0}^{n} a_kx^k\) refers to the series formed by summing the terms of the sequence \(a_0, a_1x, a_2x^2, ..., a_nx^n\). Summation notation is a compact way to represent these long additions and provides a systematic method for calculating the sum.

Whether dealing with finite series (like the sum of the squares of the first 100 natural numbers) or infinite series, the concept of adding sequence terms remains central. Through sequences, we understand the arrangement of numbers, and through series, we uncover their aggregate behavior and sums.