Index shift - sequences and series, Mathematics

Assignment Help:

Index Shift - Sequences and Series

The main idea behind index shifts is to start a series at a dissimilar value for whatever the reason (and yes, there are legitimate reasons for doing that).

Consider the following series,

992_Index Shift - Sequences and Series 1.png

Assume that for some reason we wanted to start this series at n = 0 , but we did not wish to change the value of the series. The meaning of this is that we can't just change the n = 2 to n = 0 as this would add in two new terms to the series and so change its value.

Carrying out an index shift is a quite simple process to do. We'll start by describing a new index, say i, as follows,

i = n - 2

Here now, when n = 2, we will get i = 0 . Notice as well that if n = ∞ then i = ∞- 2 = ∞ , so only the lower limit will alter here. Next, we can solve this for n to get,

n = i + 2

We can now totally rewrite the series in terms of the index i in place of the index n just by plugging in our equation for n in terms of i.

717_Index Shift - Sequences and Series 2.png

To end the problem out we'll remind that the letter we employed for the index doesn't matter and thus we'll change the final i back into an n to get,

1999_Index Shift - Sequences and Series 3.png

To induce you that these really are similar summation let us write out the first couple of terms for each one of them,

1289_Index Shift - Sequences and Series 4.png

Thus, sure enough the two series do have exactly similar terms.

In fact there is an easier way to do an index shift. The method described above is the technically right way of doing an index shift. Though, notice in the above instance we decreased the initial value of the index by 2 and all the n's in the series terms increased by 2 also. This will all time work in this way.  If we decrease the initial value of the index by a set amount as compared to all the other n's in the series term will increase by similar amount. Similarly, if we increase the initial value of the index by a set amount, after that all the n's in the series term will decrease by similar amount.


Related Discussions:- Index shift - sequences and series

Decision-making under conditions of certainty, Decision-Making Under Condit...

Decision-Making Under Conditions of Certainty Conditions of certainty tend to be rare, especially when significant decisions are involved. Under conditions of certainty, decis

Introduction to multiplication and division, INTRODUCTION :  When a Class ...

INTRODUCTION :  When a Class 5 child was given the problem 'If I paid Rs. 60 for 30 pencil boxes, how much did b pencil box cost?', he said it would be 60 x 30 = 1800. This

My homework, Paulina played 3 soccer games on Saturday she drank I juice bo...

Paulina played 3 soccer games on Saturday she drank I juice box during each soccer game how many juice boxes did she drank

Power of iota, The next topic that we desire to discuss here is powers of i...

The next topic that we desire to discuss here is powers of i. Let's just take a look at what occurring while we start looking at many powers of i . i 1 = i

How many pounds should nicole put in every basket, Nicole is forming 20 gif...

Nicole is forming 20 gift baskets. She has 15 pounds of chocolates to distribute equally between the baskets. If each basket gets the similar amount of chocolates, how many pounds

Rolles theorem, Rolle's Theorem  Assume f(x) is a function which satis...

Rolle's Theorem  Assume f(x) is a function which satisfies all of the following. 1. f(x) is continuous in the closed interval [a,b]. 2. f(x) is differentiable in the ope

Utilizes the infinite definition of the limit to prove limit, Utilizes the ...

Utilizes the definition of the limit to prove the given limit. Solution Let M > 0 be any number and we'll have to choose a δ > 0 so that, 1/ x 2   > M

One of these food groups, In a collection of 30 dissimilar birds, 15 eat wo...

In a collection of 30 dissimilar birds, 15 eat worms, 18 eat fruit, and 12 eat seeds. Accurately 8 eat worms and seeds, 8 eat worms and fruit, 7 eat fruit and seeds, and 4 eat each

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd