Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves.

These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well$-$defined limit$.$

Objectives:

This assignment will assess your mastery of the following objectives:

Evaluate complex expressions that use multiple arithmetic operators.

Declare variables for specific purposes to store the values and cast their data types.

Use string concatenation to produce the informative output for statistical analysis.

Involve redundant tasks by using for loops according to the patterns.

Design parameterized methods to accept parameters and return a value from the methods as a result

Use the Math class to compute values using predefined common mathematical functions.

Follow prescribed conventions for spacing, indentation, naming methods, and header comments.

Rules and Explanations:

You will receive a grade of zero for using any built$-$in classes$/$methods beside Math methods

Pi Constant $(\backslash$pi $)$:

Pi is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to $3\mathrm{.}14159.$ The number $\backslash$pi appears in many formular across mathematics and physics. Your task is to implement the method pi that accept an integer $($int data type$).$ pi$($n$)$ should follow the Madhava of Sangamagrama formula and return a value based on the terms we provided in the summation. We assume that the input n will always a positive value. The method should NOT print anything to console. Here is the examples:

n pi$($n$)$

$13\mathrm{.}4641016151377544$

$23\mathrm{.}0792014356780038$

$33\mathrm{.}156181471569954$

$43\mathrm{.}1378528915956805$

$53\mathrm{.}1426047456630846$

$......$

Note: we identify the first term of the summation when i $=0$ and n $=1$

Euler Constant $()$:

The number $,$ also known as Euler's number, is a mathematical constant approximately equal to $2\mathrm{.}71828$ that can be characterized in many ways. It is the base of natural logarithms. Your task is to implement the method euler that accept an integer $($int data type$).$ euler$($n$)$ should return a value based on the terms we provided in the summation. We assume that the input n will always a positive value. The method should NOT print anything to console. Here is the examples:

$$

n euler$($n$)$

$11\mathrm{.}0$

$22\mathrm{.}0$

$32\mathrm{.}5$

$42\mathrm{.}6666666666666665$

$52\mathrm{.}708333333333333$

$......$

Note: we identify the first term of the summation when i $=0$ and n $=1$

Main Program:

Use the for loop the iterate all terms and print constant values from:

$1$ to $30$ for the pi method

$1$ to $18$ for the euler method

Here is the expected output: $(15$ digits after decimal point for each value$)$