Finding the sum of different types of series:

  1. The sum of a finite arithmetic progression:
    [
    S_n = \frac{n}{2} (a_1 + a_n)
    ]
    where ( S_n ) is the sum of the first ( n ) terms, ( a_1 ) is the first term, ( a_n ) is the last term, ( n ) is the number of terms.
  2. Sum of an infinite arithmetic progression:
    An infinite arithmetic progression does not have a finite sum unless the difference is zero.
  3. The sum of a finite geometric progression:
    [
    S_n = a_1 \frac{1 – r^n}{1 – r} \quad (r \neq 1)
    ]
    where ( S_n ) is the sum of the first ( n ) terms, ( a_1 ) is the first term, ( r ) is the denominator of the progression.
  4. The sum of an infinite geometric progression:
    [
    S = \frac{a_1}{1 – r} \quad (|r| < 1)
    ]
    where ( S ) is the sum, ( a_1 ) is the first term, ( r ) is the denominator of the progression.
  5. The sum of a series of natural numbers:
    [
    S_n = \frac{n(n + 1)}{2}
    ]
    where ( S_n ) is the sum of the first ( n ) natural numbers.
  6. Sum of the squares of the first ( n ) natural numbers:
    [
    S_n = \frac{n(n + 1)(2n + 1)}{6}
    ]
  7. Sum of cubes of first ( n ) natural numbers:
    [
    S_n = \left( \frac{n(n + 1)}{2} \right)^2
    ]

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