Task 4 [Computing Pl via Leibniz series]This task is to help you use for-loops for mathematical expressions in code.It is an irrational number - that means it cannot be expressed as a fraction. Ever wonder how we compute ?One of the pioneers of the mechanical calculator, Gottfried Wilhelm von Leibniz, proved that could beexpressed with the following infinite series:1-1 1 15 7 9 11 13 15 17 191 1 1 1 1 1++3We can rewrite Leibniz's series in a more compact form asfollows:π = 4 xNotice the alternating plus/minus sign in the series. We call each of the fractions in the series a "term." Tocompute requires the summing of an infinite number of terms; however, in practice, this is not possible.Instead, we can only approximate the true value using a finite number of terms. Using the Leibniz formula, themore terms we compute in the series, the more accurate our calculation of . Your task is to compute π upto a user-specified number of terms (i.e., up to M terms).Don't get scared of math, it is easy once you understand it.The Σ (sigma) symbol is a mathematician's way ofexpressing a "for loop" for summing up terms. Thissymbol means the following:sum=0for nMΣn=0(-1)"2n + 1Your task should work as follows:(1) Ask the user for the number of terms, M.(2) Compute pi based on Leibniz formulation.(3) At each iteration of your for-loop, print out the result(i.e., your current sum x 4)Print your results with precision 12.Compare your result with 3.14159265359.See example output below:our pi= 3.33968253968real pi= 3.14159265359n=5---- Task 4: Compute PIInput number of terms, M: 10n=0adding fraction: 1/1our pi4.00000000000real pi= 3.14159265359n=1adding fraction: -1/3our pi 2.66666666667real pi= 3.14159265359n=2adding fraction: 1/5our pi= 3.46666666667real pi= 3.14159265359n=3our pi 2.89523809524real pi= 3.14159265359n=4adding fraction: -1/7adding fraction: 1/9adding fraction: -1/11our pi 2.97604617605real pi= 3.14159265359n=6adding fraction: 1/13our pi= 3.28373848374real pi= 3.14159265359+ ...=sumrange (0,M+1):sum + [term]Here, [term] is the result of the expression inblue, evaluated using the current value of n in thefor loop. For example, if M=2, we have:TT4n=0, we haven=1, we have(-1)⁰2*0+1(-1)¹2+112*1+1(-1)²n=2, we have2*2+14+1(Look familiar? This is Leibniz's series.)3- +15 n=7adding fraction: -1/153.01707181707= 3.14159265359.. adding fraction: 1/17= 3.25236593472= 3.14159265359our pireal pin=8=our pireal pin=9 ... adding fraction: -1/19our pireal pi3.04183961893= 3.14159265359=n=10 ... adding fraction: 1/21our pi= 3.23231580941real pi = 3.14159265359Try this for M=10000. Remember, before computers, a person would have computed this by hand.Isaac Newton (not using the method above) computed up to a precision of 16 decimal places in 1665.The current record holder is Emma Haruka Iwao. She has estimated with over 31 trillion digits (and not byhand!)

To calculate the value of PI using Lebniz formulation.

Task 4 [Computing Pl via Leibniz series] This task is to help you use for-loops for mathematical expressions in code. It is an irrational number that means it cannot be expressed as a fraction. Ever wonder how we compute ? One of the pioneers of the mechanical calculator, Gottfried Wilhelm von Leibniz, proved that could be expressed with the following infinite series: 1 1 1 1 1 1 1 + + 35-7 1 1 1 + 9 11 13 15 17 19 TT 4 Notice the alternating plus/minus sign in the series. We call each of the fractions in the series a "term." To compute π requires the summing of an infinite number of terms; however, in practice, this is not possible. Instead, we can only approximate the true value using a finite number of terms. Using the Leibniz formula, the more terms we compute in the series, the more accurate our calculation of π. Your task is to compute π up to a user-specified number of terms (i.e., up to M terms). Don't get scared of math, it is easy once you understand it. The Σ (sigma) symbol is a mathematician's way of We can rewrite Leibniz's series in a more compact form as

Task 4 [Computing Pl via Leibniz series] This task is to help you use for-loops for mathematical expressions in code. It is an irrational number that means it cannot be expressed as a fraction. Ever wonder how we compute ? One of the pioneers of the mechanical calculator, Gottfried Wilhelm von Leibniz, proved that could be expressed with the following infinite series: 1 1 1 1 1 1 1 + + 35-7 1 1 1 + 9 11 13 15 17 19 TT 4 Notice the alternating plus/minus sign in the series. We call each of the fractions in the series a "term." To compute π requires the summing of an infinite number of terms; however, in practice, this is not possible. Instead, we can only approximate the true value using a finite number of terms. Using the Leibniz formula, the more terms we compute in the series, the more accurate our calculation of π. Your task is to compute π up to a user-specified number of terms (i.e., up to M terms). Don't get scared of math, it is easy once you understand it. The Σ (sigma) symbol is a mathematician's way of We can rewrite Leibniz's series in a more compact form as

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Concept explainers

Types of Loop

Loops are the elements of programming in which a part of code is repeated a particular number of times. Loop executes the series of statements many times till the conditional statement becomes false.

Loops

Any task which is repeated more than one time is called a loop. Basically, loops can be divided into three types as while, do-while and for loop. There are so many programming languages like C, C++, JAVA, PYTHON, and many more where looping statements can be used for repetitive execution.

While Loop

Loop is a feature in the programming language. It helps us to execute a set of instructions regularly. The block of code executes until some conditions provided within that Loop are true.

Question

Task 4 [Computing Pl via Leibniz series]
This task is to help you use for-loops for mathematical expressions in code.
It is an irrational number - that means it cannot be expressed as a fraction. Ever wonder how we compute ?
One of the pioneers of the mechanical calculator, Gottfried Wilhelm von Leibniz, proved that could be
expressed with the following infinite series:
1-
1 1 1
5 7 9 11 13 15 17 19
1 1 1 1 1 1
+
+
3
We can rewrite Leibniz's series in a more compact form as
follows:
π = 4 x
Notice the alternating plus/minus sign in the series. We call each of the fractions in the series a "term." To
compute requires the summing of an infinite number of terms; however, in practice, this is not possible.
Instead, we can only approximate the true value using a finite number of terms. Using the Leibniz formula, the
more terms we compute in the series, the more accurate our calculation of . Your task is to compute π up
to a user-specified number of terms (i.e., up to M terms).
Don't get scared of math, it is easy once you understand it.
The Σ (sigma) symbol is a mathematician's way of
expressing a "for loop" for summing up terms. This
symbol means the following:
sum=0
for n
M
Σ
n=0
(-1)"
2n + 1
Your task should work as follows:
(1) Ask the user for the number of terms, M.
(2) Compute pi based on Leibniz formulation.
(3) At each iteration of your for-loop, print out the result
(i.e., your current sum x 4)
Print your results with precision 12.
Compare your result with 3.14159265359.
See example output below:
our pi= 3.33968253968
real pi= 3.14159265359
n=5
---- Task 4: Compute PI
Input number of terms, M: 10
n=0
adding fraction: 1/1
our pi4.00000000000
real pi= 3.14159265359
n=1
adding fraction: -1/3
our pi 2.66666666667
real pi= 3.14159265359
n=2
adding fraction: 1/5
our pi= 3.46666666667
real pi= 3.14159265359
n=3
our pi 2.89523809524
real pi= 3.14159265359
n=4
adding fraction: -1/7
adding fraction: 1/9
adding fraction: -1/11
our pi 2.97604617605
real pi= 3.14159265359
n=6
adding fraction: 1/13
our pi= 3.28373848374
real pi= 3.14159265359
+ ...=
sum
range (0,M+1):
sum + [term]
Here, [term] is the result of the expression in
blue, evaluated using the current value of n in the
for loop. For example, if M=2, we have:
TT
4
n=0, we have
n=1, we have
(-1)⁰
2*0+1
(-1)¹
2+1
1
2*1+1
(-1)²
n=2, we have
2*2+1
4+1
(Look familiar? This is Leibniz's series.)
3
- +
1
5

n=7
adding fraction: -1/15
3.01707181707
= 3.14159265359
.. adding fraction: 1/17
= 3.25236593472
= 3.14159265359
our pi
real pi
n=8
=
our pi
real pi
n=9 ... adding fraction: -1/19
our pi
real pi
3.04183961893
= 3.14159265359
=
n=10 ... adding fraction: 1/21
our pi
= 3.23231580941
real pi = 3.14159265359
Try this for M=10000. Remember, before computers, a person would have computed this by hand.
Isaac Newton (not using the method above) computed up to a precision of 16 decimal places in 1665.
The current record holder is Emma Haruka Iwao. She has estimated with over 31 trillion digits (and not by
hand!)

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