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Lesson Topics Discussion Quiz: Class Homework

Example1

Title: Calculus

Grade Lesson s6-p2

Explanation: The best way to understand SAT-4 is by looking at some examples. Take turns and read each example for easy understanding.

Examples

Topics → Definition Example1 Example2

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

Step: 1

The given series is:

\$ \sum_{n=1}^\infty (2^(n-1))/n \$

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

⇒ \$\sum_{n=1}^\infty (2^n \times 2^-1) / n\$

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

⇒ \$ 2^(-1) \sum_{n=1}^\infty (2^n) / (n) \$

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

⇒ \$1/2 \sum_{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: 2

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.