sum_(n=0)^oo e^(-3n) Use the Root Test to determine the convergence or divergence of the series.

To apply the Root test on a series  sum a_n , we determine the limit as:
lim_(n-gtoo) root(n)(|a_n|)= L
or
lim_(n-gtoo) |a_n|^(1/n)= L
Then, we follow the conditions:
a) Llt1 then the series is absolutely convergent.
b) Lgt1 then the series is divergent.
c) L=1 or does not exist  then the test is inconclusive. The series may be divergent, conditionally convergent, or absolutely convergent.
To apply the Root Test to determine the convergence or divergence of the series sum_(n=0)^oo e^(-3n) , we  let a_n = e^(-3n) .
Apply Law of Exponent: x^(-n) = 1/x^n . 
a_ n = 1/e^(3n).
Applying the Root test, we set-up the limit as:
lim_(n-gtoo) |1/e^(3n)|^(1/n) =lim_(n-gtoo) (1/e^(3n))^(1/n)
 Apply the Law of Exponents: (x/y)^n = x^n/y^n and (x^n)^m= x^(n*m) .
lim_(n-gtoo) (1/e^(3n))^(1/n) =lim_(n-gtoo) 1^(1/n)/(e^(3n))^(1/n)
                            =lim_(n-gtoo) 1^(1/n)/e^(3n*1/n)
                          =lim_(n-gtoo) 1^(1/n)/e^((3n)/n)     
                          =lim_(n-gtoo) 1^(1/n)/e^3        
Apply the limit property: lim_(x-gta)[(f(x))/(g(x))] =(lim_(x-gta) f(x))/(lim_(x-gta) g(x)) .
lim_(n-gtoo) 1^(1/n)/e^3 =(lim_(n-gtoo) 1^(1/n))/(lim_(n-gtoo)e^3 )
                  = 1^(1/oo) /e^3
                  =1^0/e^3
                     =1/e^3 or 0.0498 (approximated value)
The limit value L = 1/e^3 or 0.0498  satisfies the condition: Llt1 since 0.0498lt1.
Thus, the series sum_(n=0)^oo e^(-3n)  is absolutely convergent.  

Comments

Post a Comment

Popular posts from this blog

In what ways might RFID technology be used to serve customers better? What problems might arise? Do you think that the technology might be valuable when implanted in animals or people?

RFID stands for Radio Frequency Identification. It is a type of wireless technology and basically has three components to it: a tag, a reader, and a computer system. The tag consists of a microchip and radio antenna. The microchip in the tag contains essential information about a product or item. To transmit this information to a reader, the tag uses radio signals. After picking up the signals, the reader delivers the information to a computer system. From the computer system, companies can easily track the kinds of products consumers like to buy. This allows companies to position advertisements, sales, and product placements in stores according to their customer's preferences. Also, using RFID technology increases productivity: it is much easier to use, is more accurate, and less error-prone than traditional bar-coding. Thus, increased productivity leads to greater profits for companies. As for consumers, the benefits of RFID technology are numerous. For example, it minimizes wait...
Read more

Calculus: Early Transcendentals, Chapter 9, 9.3, Section 9.3, Problem 18

This differential equation can be solved by the method of separating variables. Multiply both sides by dt and divide both sides of the equation by L^2 . This will bring all terms depending on t to the right side and all terms depending on L to the left side: (dL)/L^2 = kln(t)dt Now both sides can be integrated. On the left side, we have an integral of a power function: int (dL)/L^2 = int L^(-2) dL = -1L^(-1) = -1/L (The constant of integration can be omitted for now.) On the right side, the integral of ln(t) equals int ln(t)dt = tln(t) - t (This can be verified by taking the derivative using the product rule and the fact that (ln(t))' = 1/t : (tln(t) - t)' = ln(t) + t/t - 1 = ln(t)) So the after the integration the equation becomes -1/L = k(tln(t) - t) + C The constant of integration C is added to the right side here. Since the initial condition is given, L(1) = -1, C can be found now: for t = 1, ln(t) = ln(1) = 0, and -1/(-1) = k(1*0 - 1) + C = -k + C 1 = -k + C, so from here...
Read more

Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 24

a.) Suppose that $\displaystyle G(x) = 4x^2 - x^3$, find $G'(a)$ and use it to find an equation of the tangent lines to the curve $\displaystyle y = 4x^2 - x^3$ at the point $(2,8)$ and $(3, 9)$ Using the definition of the derivative of a function $G$ at a number $a$, denoted by $G'(a)$, is $\qquad \displaystyle \qquad G'(a) = \lim_{h \to 0} \frac{G(a + h) - G(a)}{h}$ We have, $ \begin{equation} \begin{aligned} \qquad G'(a) =& \lim_{h \to 0} \frac{4(a + h)^2 - (a + h)^3 - (4a^2 - a^3)}{h} && \text{Substitute $G(a + h)$ and $G(a)$}\\ \\ \qquad G'(a) =& \lim_{h \to 0} \frac{4(a^2 + 2ah + h^2) - (a^3 + 3a^2 h + 3ah^2 + h^3) - 4a^2 + a^3}{h} && \text{Expand the equation}\\ \\ \qquad G'(a) =& \lim_{h \to 0} \frac{\cancel{4a^2} + 8ah + 4h^2 - \cancel{a^3} - 3a^2h - 3ah^2 - h^3 - \cancel{4a^2} + \cancel{a^3}}{h} && \text{Combine like terms}\\ \\ \qquad G'(a) =& \lim_{h \to 0} \frac{8ah + 4h^2 - 3a^2h - 3ah^2 - h^3}{h} ...
Read more