SaiBook.us SAT-2

Lesson Example Discussion Quiz: Class Homework

Example

Title: Infinite sequence and series

Grade: 1400-a Lesson: S2-L7

Explanation: The best way to understand SAT-2 is by looking at some examples. Take turns and read each example for easy understanding.

Examples:

Evaluate the infinite series: \$ \sum_{n=1}^\infty (2^(n-1) / n)\$.

Step 1a

The given series is:

\$ ∑_(n=1)^\infty (2^(n-1) / n) \$

Simplify \$2^(n-1)\$ = \$2^n 2^(-1)\$

⇒ \$∑_(n=1)^\infty (2^n * 2^-1) / n\$

Apply the constant multiplication rule: \$ ∑ c a_n = c ∑a_n\$

⇒ \$ 2^(-1) ∑_(n=1)^\infty 2^n / (n) \$

Apply exponent rule : \$a^(-1) = 1/a\$

⇒ \$1/2 ∑_(n=1)^\infty 2^n / (n)\$

Apply series ratio test: diverges

⇒ \$1/2\$ diverges

Explanation: First, simplify the given series, then apply the constant rule. After simplification, apply the exponent, and finally get the answer. Given series \$1/2\$ diverges.

Note: Series ratio test

If L < 1, the series converges absolutely.
If L > 1 or the limit diverges, the series diverges.
If L = 1, the test is inconclusive; further tests may be needed, such as the comparison test or the root test.

Step 1b

To find the sum of the series, we can use the formula for the sum of an infinite geometric series:

\$ S = a / (1 - r) \$

where a is the first term and r is the common ratio. In this case, a = 1 and \$ r = 1/2\$. Thus,

\$ S = 1 / (1 - 1/2) = 2 \$

Therefore, the sum of the given series is 2.

Explanation: After replacing r in the sum of the series formula, the sum of the given series simplifies to 2.