Question

Find the sum of the infinite series ......
1-1+1-1+1-1................ Infinite ?

Answer 1

Hey there, 
Good question asked by you.
Well , see the answer which is as follows---

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Let's first see what is indeterminate form.

Numericals like ::
0/0 , ∞ - ∞ , 0^0 , ∞^∞ ,∞/∞ etc. are some examples of indeterminate forms which answer we can't define .
This is part of the chapter of class 11 . Chapter - Limits and Derivatives 

For example ::

p*0 = 0
0/0= p

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Well , let's move on to the solutions of the question::
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Method – 1

Let’s do some arrangement in the series like::

1-1+1-1+1-1+1..................................... ∞

1+(-1+1)+(-1+1)+(-1+1).............................. ∞ = 1

In the brackets we can see that all brackets are ending up with 0 and at the end the answer is coming out to be 1.

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Method – 2

Let’s do some arrangement in the series like ::

1-1+1-1+1-1......................................... ∞

(1-1)+(1-1)+(1-1)......................................... ∞ = 0

In this we can simply see and say it is coming out to be 0.

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Method – 3

As we talked about the indeterminate form let’s see what answer we get::

1-1+1-1+1-1......................................... ∞

1-1+1-1+1-1......................................... [∞ - ∞]

The things which are in bracket is kind of an indeterminate form or in other words can be said it has no solution or it can’t be defined.

 

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Method – 4

Now let’s move to the last method and in this another chapter of Mathematics is used which is => Sequence and Series . Class – 11

Let’s assume the series to be S.

S = 1-1+1-1+1-1+1............................ ∞ ====è equation 1

S= 1-(1-1+1-1+1-1+1............................ ∞)

Now use equation 1

S=1-S

S=1/2 or 0.5

Well, from my side if I say this method is too weird but this is also a method to get the answer.

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Conclusion ------

If we see all answer we can say that the answer is coming in range between 0 to 1.

 

All methods are appropriate but as we know we always have to reach the best method of that question.

 

And the best answer or method of this question is è Method – 4

Many IIT teachers and many experts review say that this method authentically obeys rules of Mathematics unlike other method.

So , all feel this method to be comfortable .

 

Hope you like it!

Thanks!

 

 

Answer 2

I will apply the method of reduction series let's take an example suppose u have to find value of
Y=√1+(√1+(√1..................Unexpected times)
Now
√1+(√1..................= Y
By putting value of y we get
Y=√1+y
I apply same concept thing I will reduce the series this is just an example
Sq both sides we get
Y^2=1+y
Y^2-y-1= 0 so it is quadratic in y and we can find its root by quadratic formula
(-B+,-√B^2-4ac)/2a
So this is not I am solving I am giving u the feeling to reduce series
Now come to you question
If we let 1-1+1-1+1-1..............Infinite = s we get get 1-1+1-1.........=s
S=1-(1-1+1-1+1-1+1-1...........Infinite )
So here we can see the series again form when I just common the negative sign we see the series again form
We know 1-1+1-1+1-1.......=s by putting value of 1-1+1-1+1-1.......= S we get
S=1-S
2S=1
And we get S= 0.5 or 1/2
Please note that this is the exact value of the series because 1,-1,0 is not it's real value these r imaginary values and these values don't have any link with the answer because these types of series only solve by derivative concepts of limits because these values always tends to infinite so we can only solve these types of question by reducing the series only ..... If u have any doubt ask in comments