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Roy and Bhattacharya (2020) provided Bayesian characterization of infinite series, and their most important application, namely, to the Dirichlet series characterizing the (in)famous Riemann Hypothesis, revealed insights that are not in support of the most celebrated conjecture for over 150 years. In contrast with deterministic series considered by Roy and Bhattacharya (2020), in this article we take up random infinite series for our investigation. Remarkably, our method does not require any simplifying assumption. Albeit the Bayesian characterization theory for random series is no different from that for the deterministic setup, construction of effective upper bounds for partial sums, required for implementation, turns out to be a challenging undertaking in the random setup. In this article, we construct parametric and nonparametric upper bound forms for the partial sums of random infinite series and demonstrate the generality of the latter in comparison to the former. Simulation studies exhibit high accuracy and efficiency of the nonparametric bound in all the setups that we consider. Finally, exploiting the property that the summands tend to zero in the case of series convergence, we consider application of our nonparametric bound driven Bayesian method to global climate change analysis. Specifically, analyzing the global average temperature record over the years 1850--2016 and Holocene global average temperature reconstruction data 12,000 years before present, we conclude, in spite of the current global warming situation, that global climate dynamics is subject to temporary variability only, the current global warming being an instance, and long term global warming or cooling either in the past or in the future, are highly unlikely.