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  • Write the terms of the binomial series.
  • Recognize the Taylor series expansions of common functions.
  • Recognize and apply techniques to find the Taylor series for a function.
  • Use Taylor series to solve differential equations.
  • Use Taylor series to evaluate nonelementary integrals.

In the preceding section, we defined Taylor series and showed how to find the Taylor series for several common functions by explicitly calculating the coefficients of the Taylor polynomials. In this section we show how to use those Taylor series to derive Taylor series for other functions. We then present two common applications of power series. First, we show how power series can be used to solve differential equations. Second, we show how power series can be used to evaluate integrals when the antiderivative of the integrand cannot be expressed in terms of elementary functions. In one example, we consider e x 2 d x , an integral that arises frequently in probability theory.

The binomial series

Our first goal in this section is to determine the Maclaurin series for the function f ( x ) = ( 1 + x ) r for all real numbers r . The Maclaurin series for this function is known as the binomial series    . We begin by considering the simplest case: r is a nonnegative integer. We recall that, for r = 0 , 1 , 2 , 3 , 4 , f ( x ) = ( 1 + x ) r can be written as

f ( x ) = ( 1 + x ) 0 = 1 , f ( x ) = ( 1 + x ) 1 = 1 + x , f ( x ) = ( 1 + x ) 2 = 1 + 2 x + x 2 , f ( x ) = ( 1 + x ) 3 = 1 + 3 x + 3 x 2 + x 3 , f ( x ) = ( 1 + x ) 4 = 1 + 4 x + 6 x 2 + 4 x 3 + x 4 .

The expressions on the right-hand side are known as binomial expansions and the coefficients are known as binomial coefficients. More generally, for any nonnegative integer r , the binomial coefficient of x n in the binomial expansion of ( 1 + x ) r is given by

( r n ) = r ! n ! ( r n ) !

and

f ( x ) = ( 1 + x ) r = ( r 0 ) 1 + ( r 1 ) x + ( r 2 ) x 2 + ( r 3 ) x 3 + + ( r r 1 ) x r 1 + ( r r ) x r = n = 0 r ( r n ) x n .

For example, using this formula for r = 5 , we see that

f ( x ) = ( 1 + x ) 5 = ( 5 0 ) 1 + ( 5 1 ) x + ( 5 2 ) x 2 + ( 5 3 ) x 3 + ( 5 4 ) x 4 + ( 5 5 ) x 5 = 5 ! 0 ! 5 ! 1 + 5 ! 1 ! 4 ! x + 5 ! 2 ! 3 ! x 2 + 5 ! 3 ! 2 ! x 3 + 5 ! 4 ! 1 ! x 4 + 5 ! 5 ! 0 ! x 5 = 1 + 5 x + 10 x 2 + 10 x 3 + 5 x 4 + x 5 .

We now consider the case when the exponent r is any real number, not necessarily a nonnegative integer. If r is not a nonnegative integer, then f ( x ) = ( 1 + x ) r cannot be written as a finite polynomial. However, we can find a power series for f . Specifically, we look for the Maclaurin series for f . To do this, we find the derivatives of f and evaluate them at x = 0 .

f ( x ) = ( 1 + x ) r f ( 0 ) = 1 f ( x ) = r ( 1 + x ) r 1 f ( 0 ) = r f ( x ) = r ( r 1 ) ( 1 + x ) r 2 f ( 0 ) = r ( r 1 ) f ( x ) = r ( r 1 ) ( r 2 ) ( 1 + x ) r 3 f ( 0 ) = r ( r 1 ) ( r 2 ) f ( n ) ( x ) = r ( r 1 ) ( r 2 ) ( r n + 1 ) ( 1 + x ) r n f ( n ) ( 0 ) = r ( r 1 ) ( r 2 ) ( r n + 1 )

We conclude that the coefficients in the binomial series are given by

f ( n ) ( 0 ) n ! = r ( r 1 ) ( r 2 ) ( r n + 1 ) n ! .

We note that if r is a nonnegative integer, then the ( r + 1 ) st derivative f ( r + 1 ) is the zero function, and the series terminates. In addition, if r is a nonnegative integer, then [link] for the coefficients agrees with [link] for the coefficients, and the formula for the binomial series agrees with [link] for the finite binomial expansion. More generally, to denote the binomial coefficients for any real number r , we define

( r n ) = r ( r 1 ) ( r 2 ) ( r n + 1 ) n ! .

With this notation, we can write the binomial series for ( 1 + x ) r as

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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