In Exercises 43–54, express each sum using summation notation. Us...

Question

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 4+4^2/2+4^3/3+⋯+ 4^n/n

Accepted Answer

Hello. Today we are asked to represent the following some using some nation notation and we're going to represent the index and lower limit of the summation as K and one. So what we are given is seven plus seven squared over two plus seven cubed over three plus all of the terms to the end of term seven to the power of N over N. So the last term given to us is called the term. And this is supposed to be a term that represents any term in the given some. Now we want to go ahead and check this and in order to check this we can go ahead and let seven to the power of N over N equal to the first term of the sum which is seven. And since we're letting it equal to the first term that means we're gonna allow N two equal to one. So replacing end with one is going to give us seven to the power of 1/ equal to 77 to the power of one is just going to give us seven and 7/1 gives us seven. And what we are left with is the statement seven is equal to seven which is a true statement. So we at least know that the term accurately represents the first term. But now we want to rewrite this in a form that accurately represents any term in the given some. And in order to do that we can allow N two equal to K. And when we let N equal to K we get the some seven plus seven squared over two plus seven cubed over three plus all of the terms to the cave term, seven to the power of K over K. And this is going to be a term that accurately represents any term in the given some. So our pattern for the summation notation is going to be seven to the power of K over K. Now let's go ahead and figure out the index and lower limit. Since we want to represent the index As K&1, that means we're gonna be plugging in all the values of K starting with one and going all the way to the final value. Now, since we were given an end term, that means that the final value has to be a generic end. Since N doesn't have a specific value to it. With that being said, the lower index or the lower limit is going to be K is equal to one and the upper limit is going to be K is equal to N. Since n is the final term in our given some. Since we have the lower and upper limit and our pattern, let's go ahead and put this all together in the summation notation and we first start by drawing our sigma symbol on the bottom of the sigma is where lower limit goes. And that's gonna be K is equal to one. Since that's the first term of plugging in and for upper limit we're gonna be putting the final value which is going to be end since that represents the 10th term of the sum. And finally we put the pattern to the right of the sigma notation and that's going to give us seven to the power of K over K. And this is going to be the summation notation of the given some. And with that being said, the answer to this problem is going to be a So I hope this video helps you in understanding how to write summation notation. And I'll go ahead and see you all in the next video.

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 4+4^2/2+4^3/3+⋯+ 4^n/n

Key Concepts

Summation Notation

Index of Summation

Series and Sequences

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