Write the following series in sigma notation: (1) \( -3+4+11+18+\ldots+137 \) (3) \( 6+2+\frac{2}{3}+\frac{2}{9}+\ldots \) (to infinity)

For the first series, you can observe that each term increases by 7, 7, and so on. Starting from -3, you can express the \( n \)-th term as \( a_n = -3 + 7(n-1) \). Therefore, you can write the series in sigma notation as \( \sum_{n=1}^{21} (-3 + 7(n-1)) \), with 21 being the number of terms required to sum up to 137. For the second series, it looks like a geometric series where the first term is 6 and the common ratio is \( \frac{1}{3} \). The series can be expressed as \( \sum_{n=0}^{\infty} 6 \left(\frac{1}{3}\right)^n \), capturing all the terms that diminish towards zero as \( n \) goes to infinity.